Particulate flow simulation using Lattice Boltzmann Method: A rheological study

This work deals with calculating rheological properties of a suspension of particles in a fluid. A suspension of mono- and poly-disperse circular particles in shear flow is studied using two different methods for application of shear force: (a) by placing parallel walls at the top and bottom of the domain which are moving in opposite directions with the same velocity, and (b) using the Lees–Edwards boundary condition. The system which starts moving from rest, is allowed to reach a statistically steady state. Rheological properties namely, bulks shear stress, effective viscosity and normal stress difference of the suspension at different particle-based Reynolds numbers and different mean particle area fractions are calculated. Furthermore, the effect of size distribution on the relative effective viscosity of the suspension is investigated. Comparison of the present results with empirical formulations found in the literature shows reasonable agreement.     

Dynamic simulation of rod-like and plate-like particle dispersed systems

A particle simulation method (PSM) is presented to simulate the dynamics of rod-like and plate-like particle dispersed systems. In this method, the particle is modeled with arrays of spheres connected by three types of springs. The motion of particles in flow is followed by solving the translational and rotational equations of motion for each constituent sphere. The mobility matrix for each particle is calculated to obtain the hydrodynamic force and torque exerted on each sphere. For the hydrodynamic interaction among particles, the near-field lubrication force is considered. The method was applied to the simulation of the transient behavior of particles in a shear flow by dispersing them into a cell with periodic boundaries. In semi-dilute to concentrated systems, the overshoot of viscosity was observed for rigid rod-like particle dispersed systems, but not for flexible ones. This was due to the transient change of the microstructure from the flow-directional orientation to the planar one of particles. The normal stress appeared in the flexible particle dispersed systems because of the deformation of particles. In the rectangular plate-like particle dispersed system, the planar orientation of particles was observed and furthermore the orientation of the major axis of particles in the shear direction appeared.     

Combining Lees–Edwards boundary conditions with smoothed profile-lattice Boltzmann methods to introduce shear into particle suspensions

Using walls to introduce shear into a domain causes wall effects in the calculation of rheological properties of suspensions. Employing Lees–Edwards boundary conditions as an alternative method, removes these effects. Earlier methods of solid–fluid interactions in the framework of lattice Boltzmann method, such as Ladd and ALD methods, violate conservation law of the translational and rotational momentum (Galilean invariance). In the present study, Lees–Edwards boundary condition has been combined with smoothed profile method (SPM) intending to eliminate Galilean invariance errors. The combined method is validated by allowing a particle to cross a Lees–Edwards boundary. Moreover, third-order interpolation is used for particle distribution functions leaving the domain in the velocity gradient direction to eliminate bumps in the angular velocity of the particle when crossing the Lees–Edwards boundary. As another test case, two interacting circular cylinders placed in a sheared domain using Lees–Edwards boundary condition. Comparing results with the ones presented in the literature shows good agreement.     

Flow of Blood in Eye Vessels with Allowance for Aggregation

A nonlinear mathematical model of blood flow in a vessel has been constructed where blood is considered as a suspension with aggregating and deformable particles and vessel walls possess elasticity according to the Hooke law. The blood viscosity, which is dependent on the gradient of the rate of shear on the vessel wall, becomes a function of the time in the case of pulsatory motion. The cases of normal blood flow in eye vessels and of that in diabetes mellitus have been considered as examples. It has been shown that in the latter case the contribution of the aggregability of erythrocytes to the value of the viscosity of blood substantially increases.     

Lubrication approximation in completed double layer boundary element method

This paper reports on the results of the numerical simulation of the motion of solid spherical particles in shear Stokes flows. Using the completed double layer boundary element method (CDLBEM) via distributed computing under Parallel Virtual Machine (PVM), the effective viscosity of suspension has been calculated for a finite number of spheres in a cubic array, or in a random configuration. In the simulation presented here, the short range interactions via lubrication forces are also taken into account, via the range completer in the formulation, whenever the gap between two neighbouring particles is closer than a critical gap. The results for particles in a simple cubic array agree with the results of Nunan and Keller (1984) and Stoksian Dynamics of Brady et al. (1988). To evaluate the lubrication forces between particles in a random configuration, a critical gap of 0.2 of particle's radius is suggested and the results are tested against the experimental data of Thomas (1965) and empirical equation of Krieger-Dougherty (Krieger, 1972). Finally, the quasi-steady trajectories are obtained for time-varying configuration of 125 particles.     

Flexible discretization technique for DEM-CFD simulations including thin walls

The discrete element method (DEM) coupled with computational fluid dynamics (CFD) method is regarded as a standard approach for a simulation of a gas-solid mixture system. In the DEM-CFD method, the local volume average technique is employed, and hence the fluid motion is calculated based on the void fraction. Although the accuracy of the DEM-CFD method has been improved through lots of studies, inflexibility may become a problem due to the local volume average technique. Specifically, calculations of a gas-solid flow involving thin walls is substantially impossible even by the improved DEM-CFD method. This is because the thin wall cannot be represented when its thickness becomes as large as one grid size due to usage of the local volume average technique. In order to solve this problem, a flexible discretization technique is newly proposed, where the signed distance function and the immersed boundary method are introduced into the dual grid model. In this technique, two kinds of grids are used to calculate the void fraction and the fluid flow. Thus, this technique makes it possible to simulate a gas-solid flow involving a thin wall. Verification and validation tests are performed to show the adequacy of this technique. Through this study, the proposed technique is illustrated to reproduce the exact solution and experimental results in the gas-solid flow involving the thin wall. Consequently, the proposed technique is shown to yield reasonable results in gas-solid flows involving the thin walls.     

Accurate treatment of lubrication forces between rigid spheres in viscous fluids using a traction-corrected boundary element method

Traditional boundary element methods cannot accurately resolve lubrication forces in the interstitial regions between nearly touching particles in viscous flows. In many cases, the interstitial tractions are underestimated and the relative particle velocities are overestimated resulting in significant errors in predicting particle trajectories. In order to accurately treat the lubrication forces between nearly touching particles, a traction-corrected boundary element method (TC-BEM) for multiple particles is developed by combining the analytical asymptotic solution for the tractions in the interstitial regions with the boundary element method. An adaptive meshing algorithm is developed to provide appropriate meshes on surfaces of particles with close interactions. The numerical method also employs an efficient parallelization scheme to make possible prediction of long-time behavior of particles suspended in viscous flow fields. The results of the TC-BEM are benchmarked by comparisons to analytical results for two particles in a linear shear flow and by considering the reversibility of three particles in a circular Couette flow. It is shown that the TC-BEM is able to correctly resolve the lubrication forces between nearly touching particles, thus enabling the accurate analysis of particles suspended in nonlinear shear flows.     

Interception of two spherical particles with arbitrary radii in simple shear flow

Summary The interception of two force-free and torque-free spherical particles with arbitrary radii freely suspended in simple shear flow is investigated in the limit of vanishing Reynolds number. At any instant, the flow is computed in a frame of reference with origin at the center of one particle using a cylindrical polar coordinate system whose axis of revolution passes through the center of the second particle. The problem is formulated as a decoupled system of integral equations for the zeroth, first, and second Fourier coefficients of the boundary traction with respect to the meridional angle. The derived integral equations are solved with high accuracy using a boundary element method that features adaptive discretization and automatic time-step adjustment according to the inter-particle gap. The results illustrate particle trajectories and describe the particle rotation and evolution of the stress tensor during the interception. The particle interaction is found to always cause a positive shift in the rotational phase angle due to the rolling motion at close contact. As the gap between two particles tends to zero, the shear stress diverges even though the net force and torque exerted on each particle remain zero, independent of the particle relative radius. A frictional force for rough surfaces and small gaps eliminates the slip velocity and promotes the rolling motion.     

Two-dimensional viscous flow of a suspension due to particle settling in a vertical vessel

Summary This paper investigates the influence of the viscosity of the suspension together with wall friction on sedimentation in vessels with vertical walls. The settling process is described by four basic equations, i.e. two continuity and two momentum equations. Assuming a very dilute suspension, an asymptotic expansion for small particle concentration is carried out. To first order, the well-known one-dimensional solution is recovered. The motion of the liquid, however, is a second-order effect governed by the biharmonic equation. In order to find a solution satisfying the boundary conditions, harmonic expansions are used. Thus the problem is reduced to a system of linear algebraic equations. Solutions for the two-dimensional flow of the liquid and the motion of the particles are given, and the influence on the shape of the interface between the suspension and the clear liquid is discussed.With 7 Figures     

Analysis of the particle collision behavior in spiral jet milling

Collision behaviors of particles in spiral jet milling were analyzed by using a simulation. The motion of the particles was tracked by the discrete element method (DEM), and the air flow was represented by the computational fluid dynamics (CFD). The DEM was coupled with the CFD by a one-way coupling method. The simulated air flow was validated by comparing the fluid velocity field with the measured one in a model experiment. Furthermore, the air flow and particle behaviors in a spiral jet mill used commercially were analyzed by using the simulation. As a result, the particles with a region balancing between centrifugal and radial drag forces could be mainly ground by the high-speed collisions between the particles circulating near the top and bottom walls of the grinding chamber.     

Physical and computational scaling issues in lattice Boltzmann simulations of binary fluid mixtures

We describe some scaling issues that arise when using lattice Boltzmann (LB) methods to simulate binary fluid mixtures--both in the presence and absence of colloidal particles. Two types of scaling problem arise: physical and computational. Physical scaling concerns how to relate simulation parameters to those of the real world. To do this effectively requires careful physics, because (in common with other methods) LB cannot fully resolve the hierarchy of length, energy and time-scales that arise in typical flows of complex fluids. Care is needed in deciding what physics to resolve and what to leave unresolved, particularly when colloidal particles are present in one or both of two fluid phases. This influences steering of simulation parameters such as fluid viscosity and interfacial tension. When the physics is anisotropic (for example, in systems under shear) careful adaptation of the geometry of the simulation box may be needed; an example of this, relating to our study of the effect of colloidal particles on the Rayleigh-Plateau instability of a fluid cylinder, is described. The second and closely related set of scaling issues are computational in nature: how do you scale-up simulations to very large lattice sizes? The problem is acute for systems undergoing shear flow. Here one requires a set of blockwise co-moving frames to the fluid, each connected to the next by a Lees-Edwards like boundary condition. These matching planes lead to small numerical errors whose cumulative effects can become severe; strategies for minimizing such effects are discussed.     

Direct numerical simulation of two-dimensional channel flows of electro-rheological fluids

Direct numerical simulation (DNS) of electro-rheological (ER) fluid flows in two-dimensional (2D) electrode channel has been performed by adopting a combined finite element method (FEM). Hydrodynamic interactions between the particles and the fluid are described by the Navier-Stokes equations for the fluid in combination with the equations of motion for the particles, while the multi-body electrostatic interaction is represented by the point-dipole model.ER effects on the plane channel flow for a given pressure gradient have been studied by varying the Mason number and volume fraction of the particles, and interrogating the motion of the particles in views of the formation of ER chain structures, the fluid velocity profile in the channel, and the shear stress versus the shear rate. As the Mason number decreases and volume fraction increases, the tendency that particles align to form chain structures becomes stronger. The yield stress of the ER fluid increases with the electric field intensity and the particle concentration. The quadratic correlation between the yield stress and the electric field intensity has been extracted from the present direct numerical simulation. Lastly, it has been shown that the yield stress linearly increases with the volume fraction in the intermediate range.     

Slip in Quantum Fluids

In this review we describe theoretical and experimental investigations of general slip phenomena in context with the flow of the quantum liquids³He,⁴He and their mixtures at low temperatures. The phenomenon of slip is related to a boundary effect. It occurs when sufficiently dilute gases flow along the wall of an experimental cell. A fluid is said to exhibit slip when the fluid velocity at the wall is not equal to the wall’s velocity. Such a situation occurs whenever the wall reflects the fluid particles in a specular-like manner, and/or if the fluid is describable in terms of a dilute ordinary gas (classical fluid) or a dilute gas of thermal excitations (quantum fluid). The slip effect in quantum fluids is discussed theoretically on the basis of generalized Landau-Boltzmann transport equations and generalized to apply to a regime of ballistic motion of the quasiparticles in the fluid. The central result is that the transport coefficient of bulk shear viscosity, which typically enters in the Poiseuille flow resistance and the transverse acoustic impedance, has to be replaced by geometry dependent effective viscosity, which depends on the details of the interaction of the fluid particles with the cell walls. The theoretical results are compared with various experimental data obtained in different geometries and for both Bose and Fermi quantum fluids. Good agreement between experiment and theory is found particularly in the case of pure normal and superfluid³He, with discrepancies probably arising because of deficiencies in characterization of the experimental surfaces.     

A DEM-based method for predicting the wear evolution of structural boundary composed of spherical boundary elements

In the discrete element method (DEM) simulation for wear prediction, structural boundary is now represented extensively by triangular meshes with high resolution, which brings a huge computational cost. A DEM-based method for predicting the wear evolution of structural boundary has been developed for computational efficiency. The structural boundary subjected to wear is represented by the spherical boundary elements in the DEM simulation in combination with the inside triangles and fitting curved surface in wear prediction. Wear prediction is performed through a series of evolution steps. In each evolution step, the collision energies by particles at structural boundary are collected via the DEM simulation and assigned to the boundary elements. Then, the volume losses of structural boundary are predicted across each relevant boundary element. Finally, the new geometry of structural boundary in response to wear is described by moving the boundary elements along the depths of wear individually. Through converting the contact detection between structural boundary and particles into between spherical boundary elements and particles, our method greatly reduces the computational cost in the DEM simulation. Through two numerical tests, our method has been verified to be an efficient and accurate method for the wear prediction of structural boundaries with different resolutions.     

The extended finite element method for rigid particles in Stokes flow

A new method for the simulation of particulate flows, based on the extended finite element method (X‐FEM), is described. In this method, the particle surfaces need not conform to the finite element boundaries, so that moving particles can be simulated without remeshing. The near field form of the fluid flow about each particle is built into the finite element basis using a partition of unity enrichment, allowing the simple enforcement of boundary conditions and improved accuracy over other methods on a coarse mesh. We present a weak form of the equations of motion useful for the simulation of freely moving particles, and solve example problems for particles with prescribed and unknown velocities. Copyright © 2001 John Wiley & Sons, Ltd.     

Large eddy simulation of turbulent flows in external flow field using three-step FEM–BEM model

An innovative computational model is presented for the large eddy simulation (LES) modeling of multi-dimensional unsteady turbulent flow problems in external flow field. Based on the LES principles, the model uses a pressure projection method to solve the Navier–Stokes equations in transient condition. The turbulent motion is simulated by Smagorinsky sub-grid scale (SGS) eddy viscosity model. The momentum equation of the flow motion is solved using a three-step finite element method (FEM). The external flow field is simulated using a boundary element method (BEM) by solving a pressure Poisson equation that assumes the pressure as zero at the infinity. Through extracting the boundary effects on a specified finite computational domain, the model is able to solve the infinite boundary value problems. The present model is used to simulate the flows past a two-dimensional square rib and a three-dimensional cube at high Reynolds number. The simulation results are found to be reasonable and comparable with other models available in the literature even for coarse meshes.     

基于SPH方法的剪切流驱动液滴在固体表面变形运动数值模拟研究

光滑粒子流体动力学(SPH)方法是纯拉格朗日粒子方法,可以有效避免网格法在模拟大变形过程中带来的网格扭曲等缺陷,适合模拟含大变形的剪切流驱动液滴在固体表面变形运动过程。在基于CSF模型的表面张力SPH方法基础上,采用新的边界处理方式和界面法向修正方法,引入Brackbill提出的壁面附着力边界条件处理方法,得到了含壁面附着力边界条件的表面张力算法。基于新方法模拟了剪切流驱动液滴在固体表面变形运动过程并与实验结果和VOF方法模拟结果进行了对比验证。结果表明:该方法在处理壁面附着力问题时精度较高,稳定性较好,适合处理工程中剪切流驱动液滴在固体表面变形运动问题。     

Dynamic simulation of the flow of suspensions of two-dimensional particles with arbitrary shapes

A boundary-element method is implemented for simulating the motion of two-dimensional rigid particles with arbitrary shapes suspended in a viscous fluid in Stokes flow. The numerical implementation results in a system of linear equations for the components of the hydrodynamic traction over boundary elements distributed over the particle surfaces, and for the velocity of translation and angular velocity of rotation of the particles about designated centers. The linear system is solved by the method successive substitutions based on a physically motivated iterative procedure that involves decomposing the influence matrix into diagonal blocks consisting of physical particle clusters, and performing updates by matrix-vector multiplication using the inverses of the diagonal blocks. The iterations are found to converge as long as the particles are separated by a sufficiently large distance that depends on the particle shape and level of discretization. The stiffness of the governing equations due to lubrication forces developing between intercepting particles is removed by preventing the particles from approaching one another by less than a specified distance. Simulations are carried out for doubly-periodic suspensions of circular and elliptical particles in simple shear flow. The simulation algorithm is found to be successful for particles with moderate aspect ratios, and for small and moderate areal fractions up to 0.25. Higher areal fractions require long simulation times due to the large size of spontaneously forming particle clusters. The results illustrate the performance of various aspects of the boundary-element method and provide insights into the effect of the particle areal fraction and aspect ratio on the rheological properties of the suspension and on the geometrical properties of the evolving microstructure.     

A Lagrangian gradient smoothing method for solid‐flow problems using simplicial mesh

A novel Lagrangian gradient smoothing method (L‐GSM) is developed to solve “solid‐flow” (flow media with material strength) problems governed by Lagrangian form of Navier‐Stokes equations. It is a particle‐like method, similar to the smoothed particle hydrodynamics (SPH) method but without the so‐called tensile instability that exists in the SPH since its birth. The L‐GSM uses gradient smoothing technique to approximate the gradient of the field variables, based on the standard GSM that was found working well with Euler grids for general fluids. The Delaunay triangulation algorithm is adopted to update the connectivity of the particles, so that supporting neighboring particles can be determined for accurate gradient approximations. Special techniques are also devised for treatments of 3 types of boundaries: no‐slip solid boundary, free‐surface boundary, and periodical boundary. An advanced GSM operation for better consistency condition is then developed. Tensile stability condition of L‐GSM is investigated through the von Neumann stability analysis as well as numerical tests. The proposed L‐GSM is validated by using benchmarking examples of incompressible flows, including the Couette flow, Poiseuille flow, and 2D shear‐driven cavity. It is then applied to solve a practical problem of solid flows: the natural failure process of soil and the resultant soil flows. The numerical results are compared with theoretical solutions, experimental data, and other numerical results by SPH and FDM to evaluate further L‐GSM performance. It shows that the L‐GSM scheme can give a very accurate result for all these examples. Both the theoretical analysis and the numerical testing results demonstrate that the proposed L‐GSM approach restores first‐order accuracy unconditionally and does not suffer from the tensile instability. It is also shown that the L‐GSM is much more computational efficient compared with SPH, especially when a large number of particles are employed in simulation.