Particle-resolved Direct Numerical Simulation Research Articles

Is Lagrangian weight crucial in the direct forcing immersed boundary method?

Particle resolved direct numerical simulation (PR-DNS) is one of the most powerful research tools for particle laden flows. Among a few most popular PR-DNS methods, the direct forcing immersed boundary method (DF-IBM) has obtained great success and has been adopted in various simulations of rigid particulate flows. Within DF-IBM, Eulerian and Lagrangian frameworks are used to depict the continuum and dispersed phases, respectively. Interpolation between the two frameworks is accomplished through a discrete delta function. It is generally believed that a Lagrangian weight attached to each Lagrangian marker, which is distributed on a particle’s surface, needs to be carefully chosen. To be more specific, the Lagrangian weight is supposed to match the local Eulerian cell. The matching requirement is not trivial for non-uniform Eulerian mesh or irregular shaped particles. There are various methods developed to calculate the Lagrangian weight. Here, the Lagrangian weights in a few testing cases have been calculated following two intuitively “straightforward” methods. It turns out there are substantial discrepancies in the Lagrangian weights obtained from different methods. However, further numerical examples demonstrate that such discrepancies have negligible effects on the flow dynamics. So a natural question is raised: Is Lagrangian weight crucial in the direct forcing immersed boundary method? A negative answer to this question is suggested. More detailed analysis is provided in a forthcoming paper.

Beyond the effectiveness factor: Multi-step reactions with intraparticle diffusion limitations

Though common in many industrial chemical processes, multi-step reactions occurring within porous catalyst particles provide significant challenges to previously established modeling frameworks. In this work, a system of reaction-diffusion equations is solved analytically via eigenanalysis. The intraparticle concentration profiles are integrated over the particle volume to obtain effectiveness factors for each reacting species, referred to herein as a multi-step effectiveness vector (MEV). The MEV solution is tested for the case of uniform flow past a stationary catalyst particle with a state-of-the-art, multi-step reaction scheme for fluid catalytic cracking (FCC). Outputs from particle-resolved direct numerical simulation (DNS) are compared to outputs from ordinary differentials with effective reaction closures coming from the MEV solution, the classic single-step effectiveness factor (SEF) (R. Aris, “The mathematical theory of diffusion and reaction in permeable catalysts: The theory of the steady state,” Oxford University Press, 1975), and utilizing the intrinsic rates as-is. When compared to DNS, the MEV solution shows excellent agreement for the formation of all species and quantitatively captures the intraparticle concentration profiles. By contrast, the intrinsic treatment shows an over-prediction for product formation and the SEF treatment shows mixed levels of accuracy that depend upon the reaction coupling. Specifically, the SEF treatment accurately characterizes product formation if the kinetic scheme is weakly coupled but shows significant departure if stronger coupling is present. The computational cost associated with the MEV solution is associated with eigendecomposition of the Thiele matrix, an N×N matrix where N is the number of reacting components. Overall, the method is shown to be relatively easy to implement and may significantly enhance the fidelity of chemical reactor simulations that employ multi-step reaction schemes.

A direct numerical investigation of two-way interactions in a particle-laden turbulent channel flow

Understanding the two-way interactions between finite-size solid particles and a wall-bounded turbulent flow is crucial in a variety of natural and engineering applications. Previous experimental measurements and particle-resolved direct numerical simulations revealed some interesting phenomena related to particle distribution and turbulence modulation, but their in-depth analyses are largely missing. In this study, turbulent channel flows laden with neutrally buoyant finite-size spherical particles are simulated using the lattice Boltzmann method. Two particle sizes are considered, with diameters equal to 14.45 and 28.9 wall units. To understand the roles played by the particle rotation, two additional simulations with the same particle sizes but no particle rotation are also presented for comparison. Particles of both sizes are found to form clusters. Under the Stokes lubrication corrections, small particles are found to have a stronger preference to form clusters, and their clusters orientate more in the streamwise direction. As a result, small particles reduce the mean flow velocity less than large particles. Particles are also found to result in a more homogeneous distribution of turbulent kinetic energy (TKE) in the wall-normal direction, as well as a more isotropic distribution of TKE among different spatial directions. To understand these turbulence modulation phenomena, we analyse in detail the total and component-wise volume-averaged budget equations of TKE with the simulation data. This budget analysis reveals several mechanisms through which the particles modulate local and global TKE in the particle-laden turbulent channel flow.

A massively-parallel, unstructured overset method to simulate moving bodies in turbulent flows

An unstructured overset method capable of performing direct numerical simulation (DNS) and large eddy simulation (LES) of many (O(105)) moving bodies, utilizing many computational cores (O(105)), in turbulent, incompressible fluid flow is presented. Unstructured meshes are attached to bodies and placed within a fixed background domain. Body meshes are allowed to arbitrarily overlap and move throughout the domain. Within each mesh a high resolution, unstructured, non-dissipative finite volume method is used to solve for the flow field. Boundary conditions for each mesh are provided by interpolation from flow solutions on overlapping meshes. When many unstructured meshes of different resolution overlap, care is required in the connection between the different flow solutions. An interpolant is created which seeks to preserve volume conservation of flow quantities between meshes regardless of overlapping mesh differences. An implicit fractional step method is used for time advancement, requiring the calculation of a predicted fluid velocity and corrector pressure field. For the predictor step, the resulting interpolation is directly introduced into the implicit equations for the predicted flow field. For the corrected pressure field, the continuity between meshes is weakly enforced using a penalty formulation. The pressure formulation is symmetric, positive-definite and non-singular resulting in a formulation which is readily solvable using traditional iterative matrix inversion techniques. An Arbitrary Euler-Lagrangian (ALE) method coupled to a 6 degrees of freedom rigid body equation system (6-DOF) is used for body motion. For rotation, a quaternion representation is used to solve Euler's equations of rigid body motion. A linear spring damper model, which uses geometry information readily available from the overset assembly process, is used for collisions. Validation of the method for canonical flow fields is presented including assessment of order of accuracy and kinetic energy conservation properties. Particle-resolved direct numerical simulation (PR-DNS) of single particles in various flow fields are presented for validation. PR-DNS results of 50,000 spherical particles freely moving within turbulent channel flow are shown as a demonstration of the method at full scale. LES results of a marine propeller under crashback conditions are shown to demonstrate the ability to simulate highly unsteady turbulent flows over complex, moving geometries.

Implementation of pseudo-turbulence closures in an Eulerian–Eulerian two-fluid model for non-isothermal gas–solid flow

The Eulerian–Eulerian two-fluid model is widely used for computational fluid dynamics simulations of gas–solid flows. For non-isothermal flows, the averaged conservation equations solved in the two-fluid model require closures for drag, gas–solid heat transfer, pseudo-turbulent velocity fluctuations and the pseudo-turbulent heat flux (PTHF). However, the pseudo-turbulence terms are usually neglected in two-fluid simulations due to the lack of accurate correlations. With the increase in computational power, closures for these terms are now available from particle-resolved direct-numerical simulation (PR-DNS). Here, the PTHF closure as well as the heat-transfer closure (i.e.,the Nusselt number) extracted from PR-DNS are implemented in the two-fluid thermal energy equation in OpenFOAM. The implementation is validated by comparing the simulation results with the PR-DNS data for the temperature profiles. Based on the analysis of the thermal energy budget, the PTHF can have a significant contribution and neglecting it can lead to large errors.

Boundary effects on the drag coefficient and average Nusselt number of a sphere in SCW: A comparative study

A comparative study on cold supercritical water (SCW) passing over a hot sphere is carried out using the particle-resolved direct numerical simulation (PR-DNS) to examine the wall effects. Different working conditions are considered to obtain the drag coefficient (Cd) and average Nusselt number (Nu) of hot spheres in SCW whose physical properties significantly change with temperature and pressure. The effects of wall factors on Cd and Nu under two different boundary conditions, namely a stationary one for the fluidized sphere problem and a moving one for the settling sphere problem, are compared and discussed. Numerical results show that Cd and Nu of a sphere in SCW cannot fit the previous correlations for conventional fluids well. Therefore, these correlations are improved for SCW, which are good tools to understand the basic knowledge of SCW.

Particle-resolved numerical simulations of the gas–solid heat transfer in arrays of random motionless particles

Particle-resolved direct numerical simulations of non-isothermal gas–solid flows have been performed and analyzed from microscopic to macroscopic scales. The numerical configuration consists in an assembly of random motionless spherical particles exchanging heat with the surrounding moving fluid throughout the solid surface. Numerical simulations have been carried out using a Lagrangian VOF approach based on fictitious domain framework and penalty methods. The entire numerical approach (numerical solution and post-processing) has first been validated on a single particle through academic test cases of heat transfer by pure diffusion and by forced convection for which analytical solution or empirical correlations are available from the literature. Then, it has been used for simulating gas–solid heat exchanges in dense regimes, fully resolving fluid velocity and temperature evolving within random arrays of fixed particles. Three Reynolds numbers and four solid volume fractions, for unity Prandtl number, have been investigated. Two Nusselt numbers based, respectively, on the fluid temperature and on the bulk (cup-mixing) temperature have been computed and analyzed. Numerical results revealed differences between the two Nusselt numbers for a selected operating point. This outcome shows the inadequacy of the Nusselt number based on the bulk temperature to accurately reproduce the heat transfer rate when an Eulerian–Eulerian approach is used. Finally, a connection between the ratio of the two Nusselt numbers and the fluctuating fluid velocity–temperature correlation in the mean flow direction is pointed out. Based on such a Nusselt number ratio, a model is proposed for it.

Gas flow maldistribution in random packed beds of non-spherical particles – A CFD study

In this work, gas maldistribution in random packings of cylinders and Raschig rings is studied with particle-resolved direct numerical simulation (PRDNS) employing a finite volume solver for Cartesian grids and the immersed boundary method for the treatment of complex geometries. The geometry of the random packed bed has been obtained in a separate simulation based on a sequential deposition of particles in a cylindrical column and a simplified mechanics with additional radial force allowing for higher packing densities. The first part of the work focuses on comparison of the packings of cylinders and rings with respect to their ability to distribute the flow uniformly over the cross-section of the column when the gas in the form of a narrow jet enters along the column axis and impinges on the bed directly. The second part is devoted to analysis of three different inlet configurations and their influence on gas maldistribution in the packing of Raschig rings. It has been found that the random packing of rings has comparable properties of flow uniformisation to the same arrangement of full cylinders with significantly lower pressure drop and much less pronounced effect of formation of localised jets at the bed outlet. Among the considered inlet configurations the most beneficial is the one in which the gas duct is tangential to the walls of the distribution chamber.

Improvement of the Viscous Penalty Method for Particle-Resolved Simulations

A numerical study of the parameters controlling the viscous penalty method is investigated to better set up Particle-Resolved Direct Numerical Simulations (PR-DNS) of particulate flows. Based on this analysis, improvements of the methods are proposed in order to reach an almost second order convergence in space. The viscous penalty method is validated in Stokes regime by simulating a uniform flow past a fixed isolated cylinder. Moreover, it is also utilized in moderate Reynolds number regime for a uniform flow past a square configuration of cylinder and compared in terms of friction factor to the well-known Ergun correlation.

Particle velocity distributions and velocity fluctuations of non-Brownian settling particles by particle-resolved direct numerical simulation

Settling dynamics of non-Brownian particles is investigated using particle-resolved direct numerical simulations. There are two aims of this paper: first is to study the variation of particle velocity fluctuations with solid volume for a wide range of settling Reynolds number; second is to investigate the effects of solid volume fraction and settling Reynolds number on the distribution of particle velocity fluctuations after reaching the steady state. Simulations are performed in the periodic domain and the settling Reynolds number and solid volume fraction are varied from 0.5 to 400 and 0.01 to 0.2, respectively. It is observed that particle velocity fluctuations increase with increase in solid volume fraction for settling Reynolds number less than 100. Further increase in the settling Reynolds number alters the behavior of settling particles, making the particle velocity fluctuations either remain nearly constant or even decrease with the increase in solid volume fraction. For dense suspensions, it is observed by simulations that the distribution of particle velocities has Gaussian form during settling. However, for dilute and moderately dense suspensions, particle velocity distribution becomes non-Gaussian for the studied range of settling Reynolds number. At the end of paper, the physics behind the scaling of particle velocity fluctuations with solid volume fraction and the distribution of particle velocity fluctuations during settling is explained.

Effect of density ratio on velocity fluctuations in dispersed multiphase flow from simulations of finite-size particles

Velocity fluctuations in the carrier phase and dispersed phase of a dispersed multiphase flow are studied using particle-resolved direct numerical simulation. The simulations correspond to a statistically homogeneous problem with an imposed mean pressure gradient and are presented for $${Re}_{m}=20$$ and a wide range of dispersed phase volume fractions $$\left( 0.1 \le \phi \le 0.4 \right) $$ and density ratios of the dispersed phase to the carrier phase $$\left( 0.001 \le \rho _\mathrm{p}/\rho _\mathrm{f} \le 1000 \right) $$ . The velocity fluctuations in the fluid and dispersed phase at the statistically stationary state are quantified by the turbulent kinetic energy (TKE) and granular temperature, respectively. It is found that the granular temperature increases with a decrease in the density ratio and then reaches an asymptotic value. The qualitative trend of the behavior is explained by the added mass effect, but the value of the coefficient that yields quantitative agreement is non-physical. It is also shown that the TKE has a similar dependence on the density ratio for all volume fractions studied here other than $$\phi =0.1$$ . The anomalous behavior for $$\phi =0.1$$ is hypothesized to arise from the interaction of particle wakes at higher volume fractions. The study of mixture kinetic energy for different cases indicates that low-density ratio cases are less efficient in extracting energy from mean flow to fluctuations.

Fully-resolved direct numerical simulations of particle motion in a turbulent channel flow with the lattice-Boltzmann method

Particle-resolved direct numerical simulations (PR-DNS) of particle-laden turbulent flow in a channel are carried out to understand the effect of particle Stokes number (St) on their motion. The study focuses on particles which are larger than the Kolmogorov scale. The forces that impact this motion are factored into the discussion. The lattice-Boltzmann method (LBM) is employed for the simulations. The scheme employed is able to resolve the surface of the particle and a method is adopted to account for the exchange of momentum between the particles and fluid as the particles move on fixed lattices. The simulations show that particles with relatively lower St move preferentially toward the wall while those with higher St exhibit a relatively uniform concentration.

Accurate estimate of drag forces using particle-resolved direct numerical simulations

An accurate force estimate for finite-size particle simulations is proposed based on Lagrange extrapolation of third order, coupled with a Taylor interpolation of the same order, to estimate pressure and viscous constraints on the surface of particles. The main point of our approach is to upwind the interpolation support in the normal direction to the fluid/solid interface so as to use only fluid values to estimate forces. Also, detailed validations of forces are considered for estimating accuracy and convergence order of the method on various incompressible motions such as the flow around an isolated particle at various Reynolds numbers and flows across packed spheres under faced-centered cubic, random, and bidisperse arrangements.

Turbulent channel flow past a moving array of spheres

We have performed a particle-resolved direct numerical simulation of a turbulent channel flow past a moving dilute array of spherical particles. The flow shares important features with dilute vertical gas solid flow at high Stokes number, such as significant attenuation of the turbulence kinetic energy (TKE) at low particle volume fraction. The flow has been simulated by means of an overset grid method, using spherical grids around each particle overset on a background non-uniform Cartesian grid. The main focus of the present paper is on the TKE budget, which is analysed both in the fixed channel frame of reference and in the moving particle frame of reference. The overall (domain-integrated) TKE and turbulence production due to mean shear are reduced compared to unladen flow. In the fixed frame, the interfacial term, which represents production due to relative (slip) velocity, accounts for approximately 40 % of the total turbulence production in the channel. As a consequence, the total turbulence production and the overall turbulence dissipation rate remain approximately the same as in the unladen flow. However, a comparison with laminar flow past the same particle configuration reveals that significant parts of various fixed-frame statistics are due to non-turbulent structures, spatial variations that are steady in the moving particle frame. In order to obtain a clearer picture of the modification of the true turbulence and in order to reveal the rich three-dimensional (3-D) statistical structure of turbulence interacting with particles, time averaging in the moving frame of reference of the particle is used to extract the fluctuations entirely due to true turbulence. In the moving frame, the turbulence production is positive near the sides and in the wake, but negative in a region near the front of the particle. The turbulence dissipation rate and even more the dissipation rate of the 3-D mean flow attain very large values on a large part of the particle surface, up to approximately 400 and 4000 times the local turbulence dissipation rate of the unladen flow, respectively. Very close to the particle, viscous diffusion is the dominant transport term, but somewhat further away, in particular near the front and sides of the particle, pressure diffusion and also convection provide large and positive transport contributions to the moving-frame budget. A radial analysis shows that the regions around the particles draw energy from the regions further away via the surprising dominance of the pressure diffusion flux over a large range of radii. Spectra show that (very) far away from the particles all scales of the (true) turbulence are reduced. Near the particles enhancement of small scale turbulence is observed, for the streamwise component of the velocity fluctuation more than for the other components. The most important reason for turbulence reduction and anisotropy increase appears to be particle-induced non-uniformity of the mean driving force of the flow.

Prediction on drag force and heat transfer of spheroids in supercritical water: A PR-DNS study

To carry out numerical simulations on the particle-laden supercritical water (SCW), one needs to know drag coefficient (Cd) and average Nusselt number (Nu) of the solid particles in SCW according to the high temperature and pressure conditions. This study conducts particle-resolved direct numerical simulations (PR-DNS) to obtain Cd and Nu of spheroids in SCW whose physical properties significantly change with temperature and pressure. A series of case studies are designed to match different working conditions. Effects of variable physical property parameters of SCW on Cd and Nu of spheroids are discussed. Numerical results show that the variable viscosity of SCW plays a more important role in influencing on Cd of spheroids in SCW rather than density, thermal conductivity or specific heat capacity. All these four parameters could affect Nu and the largest effect comes from the specific heat capacity. New correlations of Cd and Nu for spheroids in SCW are proposed according to the numerical results which can facilitate the phase coupling in multi-phase flow modellings of SCW.

A direct comparison of particle-resolved and point-particle methods in decaying turbulence

We use particle-resolved direct numerical simulation (PR-DNS) as a model-free physics-based numerical approach to validate particle acceleration modelling in gas-solid suspensions. To isolate the effect of the particle acceleration model, we focus on point-particle direct numerical simulation (PP-DNS) of a collision-free dilute suspension with solid-phase volume fraction$\unicode[STIX]{x1D719}=0.001$in a decaying isotropic turbulent particle-laden flow. The particle diameter$d_{p}$in the suspension is chosen to be the same as the initial Kolmogorov length scale$\unicode[STIX]{x1D702}_{0}$($d_{p}/\unicode[STIX]{x1D702}_{0}=1$) in order to overlap with the regime where PP-DNS is valid. We assess the point-particle acceleration model for two different particle Stokes numbers,$St_{\unicode[STIX]{x1D702}}=1$and 100. For the high Stokes number case, the Stokes drag model for particle acceleration under-predicts the true particle acceleration. In addition, second moment quantities which play key roles in the physical evolution of the gas–solid suspension are not correctly captured. Considering finite Reynolds number corrections to the acceleration model improves the prediction of the particle acceleration probability density function and second moment statistics of the point-particle model compared with the particle-resolved simulation. We also find that accounting for the undisturbed fluid velocity in the acceleration model can be of greater importance than using the most appropriate acceleration model for a given physical problem.

Thermal effects on the sedimentation behavior of elliptical particles

We investigate thermal effects on the sedimentation behavior of elliptical particles via particle-resolved direct numerical simulation. Two scenarios of fluid-particle heat transfer in an infinitely long channel are considered: one is a cold particle settling in a hot fluid, while the other is a hot particle settling in a cold fluid. Results show that when an elliptical particle sediment in a wide channel (i.e., the block ratio is large), in addition to the two sedimentation modes reported in the literature for the particle sediments in isothermal fluids, there exist another three modes arising from thermal effects: the tumbling mode, the anomalous rolling mode and the inclined mode. Specifically, for a cold particle settling in a hot fluid, we found the tumbling mode and inclined mode, while for a hot particle settling in a cold fluid, we found the anomalous rolling mode and inclined mode. The phase diagrams of the sedimentation modes as functions of Archimedes and Grashof numbers are given. Analyses of the relationship between particle Reynolds number and Grashof number indicates that the mode presented depends on the competition between channel wall confinement, combined forced and natural convection.

Particle resolved direct numerical simulation of free settling particles for the study of effects of momentum response time on drag force

Dynamics of settling particles is studied by particle resolved direct numerical simulation in periodic domain. The main aim of this study is to explain the effects of momentum response time on the average drag force of settling process for different solid volume fractions. The response time is varied by changing the settling fluid, solid volume fraction and Reynolds number. Simulations showed that the particles with larger response times are negligibly affected by fluid. Thus, the average drag force acting on larger response times particles is nearly same as the average drag force on fixed particles. However, for particles with smaller response times, particles develop microstructures and velocity fluctuations during settling process. This behavior of low response time settling particles results in significant deviation of average drag force from the average drag force on fixed particles. At the end of paper, an improved and better drag correlation is proposed for performing mesoscopic simulations.

Verification of Heat and Mass Transfer Closures in Industrial Scale Packed Bed Reactor Simulations

Particle-resolved direct numerical simulation (PR-DNS) is known to provide an accurate detailed insight into the local flow phenomena in static particle arrays. Most PR-DNS studies in literature do not account for reactions taking place inside the porous particles. In this study, PR-DNS is performed for catalytic reactions inside the particles using the multifluid approach where all heat and mass transfer phenomena are directly resolved both inside and outside the particles. These simulation results are then used to verify existing 1D model closures from literature over a number of different reaction parameters including different reaction orders, multiple reactions and reactants, interacting reactions, and reactions involving gas volume generation/consumption inside the particle. Results clearly showed that several modifications to existing 1D model closures are required to reproduce PR-DNS results. The resulting enhanced 1D model was then used to accurately simulate steam methane reforming, which includes all of the aforementioned reaction complexities. The effect of multiple reactants was found to be the most influential in this case.

Detailed experimental and numerical investigation of fluid–particle interactions of a fixed train of spherical particles inside a square duct

In this work, a detailed comparison of velocity fields obtained from Particle Image Velocimetry (PIV) experimental measurement and Particle-Resolved Direct Numerical Simulation (PR-DNS) of flow inside a square duct around a train of five spherical particles at a Reynolds number of 100 based on the particle diameter Dp for two different particle separation distances of l/Dp (l is the center to center distance between two neighboring particles) of 1.38 and 2.19 was performed. The PR-DNS numerical results were validated with PIV experimental data. It is argued that validations using point-wise flow measurement (such as drag on particle) are potentially unreliable for simulation verification and hence, field comparisons are needed. In addition, conclusions are drawn about the interaction of flow with spherical particles in the arrangement depending on the distance over diameter ratio and drag coefficient on the spheres were calculated.