A stochastic approach for the simulation of collisions between colloidal particles at large time steps

This paper presents a new approach for the detection and treatment of colloidal particle collisions. It has been developed in the framework of Lagrangian approaches where a large number of particles is explicitly tracked. The key idea is to account for the continuous trajectories of both colliding partners during a time step that is not restricted. Unlike classical approaches which consider only the distances between a pair of particles at the beginning and at the end of each time step (or assume straight-line motion in between), we model the whole relative, and possibly diffusive, trajectory. The collision event is dealt with using the probability that the relative distance reaches a minimum threshold (equal to the sum of the two particle radii). In that sense, the present paper builds on the idea of a previous work. However, in this first work, the collision event was simulated with a simplified scheme where one of the collision partners was removed and re-inserted randomly within the simulation domain. Though usually applied, this treatment is limited to homogeneous situations. Here, an extension of the stochastic model is proposed to treat more rigorously the collision event via a suitable evaluation of the time and spatial location of the collision and an adequate calculation of subsequent particle motion. The resulting collision kernels are successfully compared to theoretical predictions in the case of particle diffusive motion. With these promising results, the feasibility of simulating the collisional regime over a whole range of particle sizes (even nanoscopic) and time steps (from a ballistic to a purely diffusive regime) with a numerical method of reasonable computational cost has been confirmed. The present approach thus appears as a good candidate for the simulation of the agglomeration phenomenon between particles also in complex non-homogeneous flows.     

颗粒群碰撞搜索及CFD-DEM耦合分域求解的推进算法研究

在采用计算流体力学-离散元耦合方法(computational fluiddynamics-discrete element method, CFD-DEM)进行固液两相耦合分析时, 颗粒计算时间步的选取直接影响到耦合计算精度和计算效率. 为此, 本文选取每个目标颗粒为研究对象, 引入插值函数计算时间步的运动位移, 构建可变空间搜索网格; 通过筛选可能碰撞颗粒建立搜索列表, 采用逆向搜索方式判断碰撞颗粒, 从而提出一种改进的DEM方法(modified discreteelement method, MDEM). 该算法在颗粒群与流体耦合计算中, 颗粒计算初始时间步选取不受颗粒碰撞时间限制, 通过自动调整和修正实现大步长, 由颗粒和流体耦合条件实时更新流体计算时间步, 使颗粒计算时间步选取过小导致计算效率低、选取过大导致颗粒碰撞漏判的问题得以解决, 为颗粒与流体耦合的数值模拟提供了行之有效的计算方法. 通过两个颗粒和多个颗粒的数值模拟, 得到的颗粒间碰撞力、碰撞位置及次数, 与理论计算结果的相对误差均低于2%, 与传统的DEM碰撞搜索算法相比, 在选取的3种计算时间步均不会影响计算精度, 且有较高的计算效率. 通过多个颗粒与流体的耦合数值模拟, 采用传统的CFD-DEM方法, 只有颗粒计算时间步选取10$^{-6}$ s或更小才能得到精确解, 而采用本文方法取10$^{-4}$ s也能够得到精确解, 避免了颗粒碰撞随时间步增大而出现的漏判问题, 且计算耗时降低了16.7%.      

考虑颗粒碰撞的多重Monte Carlo算法

从减少计算代价和改进碰撞算法出发, 提出了考虑颗粒碰撞的多重Monte Carlo算法, 它采用直接模拟Monte Carlo算法来考虑颗粒碰撞, 并与求解颗粒拉氏Langevin方程的Monte Carlo算法耦合起来, 跟踪比实际颗粒数目小得多的虚拟颗粒. 提出了时间步长选定标准、虚拟碰撞伙伴所在控制容积的判断准则、颗粒碰撞发生的判 断准则、虚拟碰撞伙伴的选择、基于随机碰撞角度的碰撞动力学, 构成了考虑颗粒碰撞的完整多重Monte Carlo算法. 对理想工况的细微颗粒流和粗重颗粒流进行了数值模拟, 颗粒碰撞率的模拟结果与理论分析解和DNS结果均符合很好, 颗粒场演变的细节信息, 如时间平均和特定时刻的颗粒数密度, 速度和颗粒湍动能等, 均与DNS结果符合很好. 数值模拟结果证明该算法不仅具有较低的计算代价, 而且能够达到足够的计算精度.     

Statistical models for predicting the effect of bidisperse particle collisions on particle velocities and stresses in homogeneous anisotropic turbulent flows

The purpose of this paper is to present and compare two statistical models for predicting the effect of collisions on particle velocities and stresses in bidisperse turbulent flows. These models start from a kinetic equation for the probability density function (PDF) of the particle velocity distribution in a homogeneous anisotropic turbulent flow. The kinetic equation describes simultaneously particle–turbulence and particle–particle interactions. The paper is focused on deriving the collision terms in the governing equations of the PDF moments. One of the collision models is based on a Grad-like expansion for the PDF of the velocity distributions of two particles. The other model stems from a Grad-like expansion for the joint fluid–particle PDF. The validity of these models is explored by comparing with Lagrangian simulations of particle tracking in uniformly sheared and isotropic turbulent flows generated by LES. Notwithstanding the fact that the fluid turbulence may be isotropic, the particle velocity fluctuations are anisotropic due to the impact of gravitational settling. Comparisons of the model predictions and the numerical simulations show encouraging agreement.     

Modelling of particle-wall collisions in confined gas-particle flows

This paper demonstrates that numerical simulations of confined particulate two-phase flows require a detailed modelling of particle—wall collisions which includes the wall surface structure and the particle shape. These effects are taken into account by “irregular bouncing” models which are based on the statistical treatment of the collision process. In this study, results obtained using various “irregular bouncing” models based on the impulse equations for a particle—wall collision are considered and compared with experimental observations. The wall roughness is simulated by assuming that the particle collides with a virtual wall which has a randomly distributed inclination with respect to the plane, smooth wall. A Gaussian distribution for this random inclination showed the best agreement with experimental results. Numerical predictions of a turbulent two—phase flow in a vertical channel, where the particle phase is treated using a Lagrangian approach, showed that the different models applied for a particle-wall collision have a strong effect on the particle velocity fluctuations and the mass flux profiles in the region of fully developed flow. The numerical simulations using the irregular bouncing models yielded considerably higher values for the particle velocity fluctuations, which also agreed better with the experimental values. This effect was most pronounced for large particles, where the distance they need to respond to the fluid flow is larger than the characteristic dimension of the confinement. On the other hand, the motion of small particles is less affected by the choice of the wall-collision model. These effects of the wall roughness on the velocity fluctuations of the dispersed phase have not been considered in previous studies using irregular bouncing models.     

Turbulent collision of inertial particles: Point-particle based,hybrid simulations and beyond

Point-particle based direct numerical simulation (PPDNS) has been a productive research tool for studying both single-particle and particle-pair statistics of inertial particles suspended in a turbulent carrier flow. Here we focus on its use in addressing particle-pair statistics relevant to the quantification of turbulent collision rate of inertial particles. PPDNS is particularly useful as the interaction of particles with small-scale (dissipative) turbulent motion of the carrier flow is mostly relevant. Furthermore, since the particle size may be much smaller than the Kolmogorov length of the background fluid turbulence, a large number of particles are needed to accumulate meaningful pair statistics. Starting from the relative simple Lagrangian tracking of so-called ghost particles, PPDNS has significantly advanced our theoretical understanding of the kinematic formulation of the turbulent geometric collision kernel by providing essential data on dynamic collision kernel, radial relative velocity, and radial distribution function. A recent extension of PPDNS is a hybrid direct numerical simulation (HDNS) approach in which the effect of local hydrodynamic interactions of particles is considered, allowing quantitative assessment of the enhancement of collision efficiency by fluid turbulence. Limitations and open issues in PPDNS and HDNS are discussed. Finally, on-going studies of turbulent collision of inertial particles using large-eddy simulations and particle-resolved simulations are briefly discussed.     

A unified gas-kinetic scheme for multiscale and multicomponent flow transport

Compressible flows exhibit a diverse set of behaviors, where individual particle transports and their collective dynamics play different roles at different scales. At the same time, the atmosphere is composed of different components that require additional degrees of freedom for representation in computational fluid dynamics. It is challenging to construct an accurate and efficient numerical algorithm to faithfully represent multiscale flow physics across different regimes. In this paper, a unified gas-kinetic scheme(UGKS) is developed to study non-equilibrium multicomponent gaseous flows. Based on the Boltzmann kinetic equation, an analytical space-time evolving solution is used to construct the discretized equations of gas dynamics directly according to cell size and scales of time steps, i.e., the so-called direct modeling method. With the variation in the ratio of the numerical time step to the local particle collision time(or the cell size to the local particle mean free path), the UGKS automatically recovers all scale-dependent flows over the given domain and provides a continuous spectrum of the gas dynamics. The performance of the proposed unified scheme is fully validated through numerical experiments.The UGKS can be a valuable tool to study multiscale and multicomponent flow physics.     

Numerical simulation of polygonal particles moving in incompressible viscous fluids

A two-dimensional coupled lattice Boltzmann immersed boundary discrete element method is introduced for the simulation of polygonal particles moving in incompressible viscous fluids. A collision model of polygonal particles is used in the discrete element method. Instead of a collision model of circular particles, the collision model used in our method can deal with particles of more complex shape and efficiently simulate the effects of shape on particle–particle and particle–wall interactions. For two particles falling under gravity, because of the edges and corners, different collision patterns for circular and polygonal particles are found in our simulations. The complex vortexes generated near the corners of polygonal particles affect the flow field and lead to a difference in particle motions between circular and polygonal particles. For multiple particles falling under gravity, the polygonal particles easily become stuck owing to their corners and edges, while circular particles slip along contact areas. The present method provides an efficient approach for understanding the effects of particle shape on the dynamics of non-circular particles in fluids.     

An empirical model predicting the viscosity of highly concentrated, bimodal dispersions with colloidal interactions

The relationship between particle size distribution and viscosity of concentrated dispersions is of great industrial importance, since it is the key to get high solids dispersions or suspensions. The problem is treated here experimentally as well as theoretically for the special case of strongly interacting colloidal particles. An empirical model based on a generalized Quemada equation is used to describe η as a function of volume fraction for mono- as well as multimodal dispersions. The pre-factor η˜ accounts for the shear rate dependence of η and does not affect the shape of the η vs φ curves. It is shown here for the first time that colloidal interactions do not show up in the maximum packing parameter and φ_max can be calculated from the particle size distribution without further knowledge of the interactions among the suspended particles. On the other hand, the exponent ɛ is controlled by the interactions among the particles. Starting from a limiting value of 2 for non-interacting either colloidal or non-colloidal particles, ɛ generally increases strongly with decreasing particle size. For a given particle system it then can be expressed as a function of the number average particle diameter. As a consequence, the viscosity of bimodal dispersions varies not only with the size ratio of large to small particles, but also depends on the absolute particle size going through a minimum as the size ratio increases. Furthermore, the well-known viscosity minimum for bimodal dispersions with volumetric mixing ratios of around 30/70 of small to large particles is shown to vanish if colloidal interactions contribute significantly. Received: 7 June 2000/Accepted: 12 February 2001     

Simulating orthokinetic heterocoagulation and cluster growth in destabilizing suspensions

Using direct numerical simulation, we investigate the coagulation behavior of non-Brownian colloidal particles as exemplified by Al₂O₃ particles. This yields the so-called capture efficiency, for which we give an analytical expression, as well as other time-dependent variables such as the cluster growth rate. Instead of neglecting or strongly approximating the hydrodynamic interactions between particles, we include hydrodynamic and non-hydrodynamic interactions in a Stokesian dynamics approach and a comprehensive modeling of the interparticle forces. The resulting parallelized simulation framework enables us to investigate the dynamics of polydisperse particle systems composed of several hundred particles at the same high level of modeling we used for a close investigation of the coagulation behavior of two unequal particles in shear flow. Appropriate cluster detection yields all the information about large destabilizing systems, which is needed for models used in flow-sheet simulations. After non-dimensionalization, the results can be generalized and applied to other systems tending to secondary coagulation.     

Measurement of inertial particle clustering and relative velocity statistics in isotropic turbulence using holographic imaging

We present the first measurements of relative velocity statistics of inertial particles in a homogeneous isotropic turbulent flow with three-dimensional holographic particle image velocimetry (holographic PIV). From the measurements we are able to obtain the radial relative velocity probability density function (PDF) conditioned on the interparticle separation distance, for distances on the order of the Kolmogorov length scale. Together with measurements of the three-dimensional radial distribution function (RDF) in our turbulence chamber, these statistics, in principle, can be used to determine interparticle collision rates via the formula derived by Sundaram and Collins (1997). In addition, we show temporal development of the RDF, which reveals the existence of an extended quasi-steady-state regime in our facility. Over this regime the measured two-particle statistics are compared to direct numerical simulations (DNS) with encouraging qualitative agreement. Statistics at the same Reynolds number but different Stokes numbers demonstrate the ability of the experiment to correctly capture the trends associated with particles of different inertia. Our results further indicate that even at moderate Stokes numbers turbulence may enhance collision rates significantly. Such experimental investigations may prove valuable in validating, guiding and refining numerical models of particle dynamics in turbulent flows.     

Flow simulations with multi-particle collision dynamics

Multi-particle collision dynamics (MPCD) is a particle based Navier-Stokes solver and in the last ten years it has been largely used to analyze mesoscopic systems where both hydrodynamics and thermal effects have to be taken into account, typical examples being colloidal suspensions and polymer solutions. Though the soundness of this approach is well documented, only a few studies present a systematic validation of the method as a Navier-Stokes solver for relatively complex flows (e.g. unsteady, non-uniform). In this study we use MPCD to simulate an unsteady periodic flow (second Stokes?? problem) and a two dimensional flow (lid-driven cavity). Quantitative comparisons with analytical and finite difference results show that MPCD is able to correctly reproduce the hydrodynamics of these systems in a wide range of numerical parameter values, allowing the applications of MPCD to the analysis of complex fluids in confined geometries such as in Lab-On-a-Chip microfluidic devices. Discrepancies for certain parameter ranges and in specific flow conditions are singled out and discussed.     

Measurement of permeability of microfluidic porous media with finite-sized colloidal tracers

We present two methods how the permeability in porous microstructures can be experimentally obtained from particle tracking velocimetry of finite-sized colloidal particles suspended in a liquid. The first method employs additional unpatterned reference channels where the liquid flow can be calculated theoretically and a relationship between the velocity of the particles and the liquid is obtained. The second method takes advantage of a time-dependent pressure drop that leads to an exponential decrease in the particle velocity inside a porous structure. From the corresponding decay time, the permeability can be calculated independently of the particle size. Both methods lead to results comparable with permeabilities derived from numerical simulations.     

Differentially weighted direct simulation Monte Carlo method for particle collision in gas-solid flows

In gas-solid flows,particle-particle interaction(typical,particle collision) is highly significant,despite the small particles fractional volume.Widely distributed polydisperse particle population is a typical characteristic during dynamic evolution of particles(e.g.,agglomeration and fragmentation) in spite of their initial monodisperse particle distribution.The conventional direct simulation Monte Carlo(DSMC)method for particle collision tracks equally weighted simulation particles,which results in high statistical noise for particle fields if there are insufficient simulation particles in less-populated regions.In this study,a new differentially weighted DSMC(DW-DSMC) method for collisions of particles with different number weight is proposed within the framework of the general Eulerian-Lagrangian models for hydrodynamics.Three schemes(mass,momentum and energy conservation) were developed to restore the numbers of simulation particle while keeping total mass,momentum or energy of the whole system unchanged respectively.A limiting case of high-inertia particle flow was numerically simulated to validate the DW-DSMC method in terms of computational precision and efficiency.The momentum conservation scheme which leads to little fluctuation around the mass and energy of the whole system performed best.Improved resolution in particle fields and dynamic behavior could be attained simultaneously using DW-DSMC,compared with the equally weighted DSMC.Meanwhile,computational cost can be largely reduced in contrast with direct numerical simulation.     

可压缩气固混合层中离散相与连续相的相互作用研究

尽管已有许多文献采用数值模拟方法研究两相流问题,但主要是集中不可压流动方面.本文采用Eul-er-Lagrange颗粒-轨道双向耦合模型对时间模式下含有固粒的二维可压缩混合层流动进行了研究.气相流场采用非定常全Navier-Stokes方程描述,并应用具有空间三阶精度的WNND(Weighted Non-Oscillatory, Contai-ning No Free Parameters and Dissipative)格式进行数值高散.固相方程采用二阶单边三点差分离散.在考虑流场对固粒作用的同时,也计及颗粒对流场的反作用.主要研究混合层大尺度涡对颗粒扩散特性的影响及颗粒对流场结构的影响问题.在对流马赫数为0.5时,研究不同Stokes数颗粒在连续流场中的扩散特性,而在对流马赫数为0.8时研究了不同Stokes数颗粒对流场小激波结构的影响.     

Numerical analysis of flow pattern of impinging liquid sprays in a cold model for cooling a hot flat plate

This paper treats the numerical analysis of two-phase mist jet flow, which is commonly adopted to cool the solidified shell in the secondary cooling zone of the continuous casting process. Flow structures of the two-phase subsonic jet impinging on a flat plate normal to flow, corresponding to the present cooling situation, are solved on the assumption that particles are perfectly elastically reflected from a surface. Again, the numerical experiments concerning mist flows composed of air and water-droplets are made in a cold model. The flow fields for both gas and particle phases strongly depend upon the particle size. When waterdroplets mixing in the mist are very small, the impinging particles travel very closely to the surface. With increasing particle size, particles are reflected from the surface in a far distance. Therefore, also, the case is analysed where a low velocity annular gas-only flow surrounding a round nozzle co-axially is present so that such idle particles may be pushed back to the surface again. This is considered to result in an improvement of the mist cooling efficiency.     

Numerical analysis of breaking waves using the moving particle semi-implicit method

The numerical method used in this study is the moving particle semi-implicit (MPS) method, which is based on particles and their interactions. The particle number density is implicitly required to be constant to satisfy incompressibility. A semi-implicit algorithm is used for two-dimensional incompressible non-viscous flow analysis. The particles whose particle number densities are below a set point are considered as on the free surface. Grids are not necessary in any calculation steps. It is estimated that most of computation time is used in generation of the list of neighboring particles in a large problem. An algorithm to enhance the computation speed is proposed. The MPS method is applied to numerical simulation of breaking waves on slopes. Two types of breaking waves, plunging and spilling breakers, are observed in the calculation results. The breaker types are classified by using the minimum angular momentum at the wave front. The surf similarity parameter which separates the types agrees well with references. Breaking waves are also calculated with a passively moving float which is modelled by particles. Artificial friction due to the disturbed motion of particles causes errors in the flow velocity distribution which is shown in comparison with the theoretical solution of a cnoidal wave. © 1998 John Wiley & Sons, Ltd.     

Nanoparticle coagulation in a planar jet via moment method

Large eddy simulations of nanoparticle coagulation in an incompressible pla- nar jet were performed.The particle is described using a moment method to approximate the particle general dynamics equations.The time-averaged results based on 3000 time steps for every case were obtained to explore the influence of the Schmidt number and the Damkohler number on the nanoparticle dynamics.The results show that the changes of Schmidt number have the influence on the number concentration of nanoparticles only when the particle diameter is less than 1 nm for the fixed gas parameters.The number concentration of particles for small particles decreases more rapidly along the flow di- rection,and the nanoparticles with larger Schmidt number have a narrower distribution along the transverse direction.The smaller nanoparticles coagulate and disperse easily, grow rapidly hence show a stronger polydispersity.The smaller coagulation time scale can enhance the particle collision and coagulation.Frequented collision and coagulation bring a great increase in particle size.The larger the Damkohler number is,the higher the particle polydispersity is.     

Efficient modelling of particle collisions using a non-linear viscoelastic contact force

In this paper the normal collision of spherical particles is investigated. The particle interaction is modelled in a macroscopic way using the Hertzian contact force with additional linear damping. The goal of the work is to develop an efficient approximate solution of sufficient accuracy for this problem which can be used in soft-sphere collision models for Discrete Element Methods and for particle transport in viscous fluids. First, by the choice of appropriate units, the number of governing parameters of the collision process is reduced to one, which is a simple combination of known material parameters as well as initial conditions. It provides a dimensionless parameter that characterizes all such collisions up to dynamic similitude. Next, a rigorous calculation of the collision time and restitution coefficient from the governing equations, in the form of a series expansion in this parameter is provided. Such a calculation based on first principles is particularly interesting from a theoretical perspective. Since the governing equations present some technical difficulties, the methods employed are also of interest from the point of view of the analytical technique. Using further approximations, compact expressions for the restitution coefficient and the collision time are then provided. These are used to implement an approximate algebraic rule for computing the desired stiffness and damping in the framework of the adaptive collision model (Kempe and Fröhlich, J. Fluid Mech. 709: 445–489, 2012). Numerical tests with binary as well as multiple particle collisions are reported to illustrate the accuracy of the proposed method and its superiority in terms of numerical efficiency.     

各向同性湍流内颗粒碰撞率的直接模拟研究

对 Re_{\lambda } 约为51均匀各向同性湍流内 St_{k}(=\tau_{p}/\tau_{k}) 为 0 ~10.0 的 有限惯性颗粒的碰撞行为进行了直接数值模拟，以研究湍流对有限惯性 颗粒碰撞的影响. 结果表明，具有一定惯性颗粒的湍流碰撞率完全不同于零惯性的轻颗粒 (St_{k}=0) 和可忽略湍流作用的重颗粒 (St{k} \to \infty) , 其变化趋势极其复杂： 在Stk为 0~1.0 之间，颗粒的碰撞率随 St 的增加而近乎线性地剧烈增长，在 Stk≈1.0 3.0(对应的StE=τp/Te≈0.5)附近，颗粒碰撞率出现两个峰值，在Stk>3.0以后，颗粒的碰撞率随惯性增 大而逐渐趋向于重颗粒极限；在峰值处，有限惯性颗粒的平均碰撞率的峰值较轻颗粒增强了 30倍左右. 为进一步分析湍流作用下颗粒碰撞率的影响因素，分别使用可能发生碰撞 的颗粒对的径向分布函数和径向相对速度来量化颗粒的局部富集效应和湍流掺混效应，表明 St_{k} \approx 1.0 时局部富集效应最为强烈，使得颗粒的碰撞率出现第1个峰值； 湍流掺混效应则随着颗粒Stk的增大而渐近增大；局部富集和湍流掺混联合作用的结果， 使得颗粒碰撞率在 St_{k} \approx 3.0 附近出现另一个峰值.