Linear flow induced in fluid particle suspension by an infinite differentially rotating disk

The steady, axisymmetric laminar flow of a homogeneous incompressible fluid with suspended particles occupying the half-infinite space over a differentially rotating rigid plane boundary is analyzed in this paper. The effect of suspended particles is described by two parametersf and τ. The mass concentration parameterf is a measure of the concentration of suspended dust particles. The interaction parameter τ is a measure of the rate at which the velocity of dust particles adjusts to changes in the fluid velocity and depends upon the size of the individual particles. Due to Ekman suction, the particle density remains no longer a constant in the boundary layer but varies with the axial coordinate ξ. Flow characteristics and density variations are studied as functions off, τ and ξ. Possible limiting cases for τ≪1 and τ≫1 which correspond to the case of fine dust and coarse dust respectively are derived and discussed.     

Numerical stochastic model for the magnetic relaxation time of the fine particle system with dipolar interactions

This paper presents the results obtained by numerical simulations, the magnetic relaxation time simulation for a fine particle system with dipolar magnetic interaction. We used a 3D simulation model for fine magnetic particles with spherical shape and lognormal distribution for their diameters. Starting from Dormann–Bessais–Fiorani model, the 3D model we used is more realistic if we consider that the particles are randomly arranged into a preset volume, following a Gaussian distribution generated with the Box–Mueller transformation.     

Simulation of suspension gravity currents with different initial aspect ratio and layout of turbidity fence

A three-dimensional (3D) numerical model, using large eddy simulation (LES), is developed for simulating the motion of suspension gravity currents. The suitable values of model parameters are determined using the existing experimental data of a two-dimensional (2D) suspension (a mixture composed of water and glass bead particles) cloud. The simulated gravity current with different initial aspect ratio (length/breadth) of the suspension is compared with the reported data of 3D laboratory experiments to investigate the effect of initial aspect ratio on the flow characteristics and the diffusion of turbidity under the presence of a turbidity fence. The comparison of simulated results of such main flow characteristics as front height, front propagation velocity and particle deposition with the experimental data reveals that the model is capable of simulating the complex behavior of the 3D suspension gravity currents to a reasonably good accuracy under complex conditions.     

Space Mapping Approach for Particle Control in Turbulent Flows

Controlling the motion of particles in turbulent flows, the paper at hand presents an efficient space–mapping approach that is based on a hierarchy of models. The approach reduces the highly complex optimization of the k-ε turbulence model for high Reynolds–number flows (fine model) to the cheaper one of the Navier–Stokes equations for smaller Reynolds–number (laminar) flows in direct numerical simulations on coarser grids (coarse model) by help of a space–map function that maps the respective coarse model control onto the desired fine model control. The numerical results are very convincing in terms of accuracy and computational effort. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Error analysis for a potential problem on locally refined grids

Summary. For the simulation of biomolecular systems in an aqueous solvent a continuum model is often used for the solvent. The accurate evaluation of the so-called solvation energy coming from the electrostatic interaction between the solute and the surrounding water molecules is the main issue in this paper. In these simulations, we deal with a potential problem with jumping coefficients and with a known boundary condition at infinity. One of the advanced ways to solve the problem is to use a multigrid method on locally refined grids around the solute molecule. In this paper, we focus on the error analysis of the numerical solution and the numerical solvation energy obtained on the locally refined grids. Based on a rigorous error analysis via a discrete approximation of the Greens function, we show how to construct the composite grid, to discretize the discontinuity of the diffusion coefficient and to interpolate the solutions at interfaces between the fine and coarse grids. The error analysis developed is confirmed by numerical experiments. Received June 25, 1998 / Revised version received July 14, 1999 / Published online June 8, 2000     

Exact solutions for two-phase colloidal-suspension transport in porous media

Two-phase transport of colloids and suspensions occurs in numerous areas of chemical, environmental, geo-, and petroleum engineering. The main effects are particle capture by the rock and altering the flux by changing the suspended and retained concentrations. Multiple mechanisms of suspended particle capture are discussed. The mathematical model for m independent particle-capture mechanisms is considered, resulting in an (m + 2) × (m + 2) system of partial differential equations. Using the stream-function as an independent variable instead of time splits the system into an (m + 1) × (m + 1) auxiliary system, containing only concentrations and one lifting hydrodynamic equation for an unknown phase saturation. Introduction of the concentration potential linked with retention concentrations yields an exact solution of the auxiliary problem. The exact formulae allow for predicting the profiles and breakthrough histories for the suspended and retained concentrations, and phase saturations. The solution shows that for small retained concentrations, the suspended concentration is in a steady-state behind the concentration front, where all the retained concentrations are proportional to the mass of suspended particles that passed via a given reservoir cross-section. The maximum penetration depths for suspended and retained particles are the same and are equal to those for a single-phase flow.     

Modeling Sediment Deposition Patterns Arising From Suddenly Released Fixed-Volume Turbulent Suspensions

Models presented in several recent papers [1–3] dealing with particle transport by, and deposition from, bottom gravity currents produced by the sudden release of dilute, well‐mixed fixed‐volume suspensions have been relatively successful in duplicating the experimentally observed long‐time, distal, areal density of the deposit on a rigid horizontal bottom. These models, however, fail in their ability to capture the experimentally observed proximal pattern of the areal density with its pronounced dip in the region initially occupied by the well‐mixed suspension and its equally pronounced local maximum at roughly the one‐third point of the total reach of the deposit. The central feature of the models employed in [1–3] is that the particles are always assumed to be vertically well‐mixed by fluid turbulence and to settle out through the bottom viscous sublayer with the Stokes settling velocity for a fluid at rest with no re‐entrainment of particles from the floor of the tank. Because this process is assumed from the outset in the models of [1–3], the numerical simulations for a fixed‐volume release will not take into account the actual experimental conditions that prevail at the time of release of a well‐mixed fixed‐volume suspension. That is, owing to the vigorous stirring that produces the well‐mixed suspension, the release volume will initially possess greater turbulent energy than does an unstirred release volume, which may only acquire turbulent energy as a result of its motion after release through various instability mechanisms. The eddy motion in the imposed fluid turbulence reduces the particle settling rates from the values that would be observed in an unstirred release volume possessing zero initial turbulent energy. We here develop a model for particle bearing gravity flows initiated by the sudden release of a fixed‐volume suspension that takes into account the initial turbulent energy of mixing in the release volume by means of a modified settling velocity that, over a time scale characteristic of turbulent energy decay, approaches the full Stokes settling velocity. Thereafter, in the flow regime, we assume that the turbulence persists and, in accord with current understanding concerning the mechanics of dense underflows, that this turbulence is most intense in the wall region at the bottom of the flow and relatively coarse and on the verge of collapse (see [22]) at the top of the flow where the density contrast is compositionally maintained. We capture this behavior by specifying a “shape function” that is based upon experimental observations and provides for vertical structure in the volume fraction of particles present in the flow. The assumption of vertically well‐mixed particle suspensions employed in [1–5] corresponds to a constant shape function equal to unity. Combining these two refinements concerning the settling velocity and vertical structure of the volume fraction of particles into the conservation law for particles and coupling this with the fluid equations for a two‐layer system, we find that our results for areal density of deposits from sudden releases of fixed‐volume suspensions are in excellent qualitative agreement with the experimentally determined areal densities of deposit as reported in [1, 3, 6]. In particular, our model does what none of the other models do in that it captures and explains the proximal depression in the areal density of deposit.     

Comparison of non-cohesive resolved and coarse grain DEM models for gas flow through particle beds

The Discrete Element Method (DEM) is a widely used approach for modelling granular systems. Currently, the number of particles which can be tractably modelled using DEM is several orders of magnitude lower than the number of particles present in common large-scale industrial systems. Practical approaches to modelling such industrial system therefore usually involve modelling over a limited domain, or with larger particle diameters and a corresponding assumption of scale invariance. These assumption are, however, problematic in systems where granular material interacts with gas flow, as the dynamics of the system depends heavily on the number of particles. This has led to a number of suggested modifications for coupled gas–grain DEM to effectively increase the number of particles being simulated. One such approach is for each simulated particle to represent a cluster of smaller particles and to re-formulate DEM based on these clusters. This, known as a representative or ‘coarse grain’ method, potentially allows the number of virtual DEM particles to be approximately the same as the real number of particles at relatively low computational cost. We summarise the current approaches to coarse grain models in the literature, with emphasis on discussion of limitations and assumptions inherent in such approaches. The effectiveness of the method is investigated for gas flow through particle beds using resolved and coarse grain models with the same effective particle numbers. The pressure drop, as well as the pre and post fluidisation characteristics in the beds are measured and compared, and the relative saving in computational cost is weighed against the effectiveness of the coarse grain approach. In general, the method is found perform reasonably well, with a considerable saving of computational time, but to deviate from empirical predictions at large coarse grain ratios.     

悬浮固粒对二维混合层流动失稳特性的影响*

本文在不可压缩二维混合层流动方程的基础之上,通过添加固粒的作用项,推导得到了修正的瑞利方程;然后用数值计算方法解其特征方程,得到了悬浮固粒的质量密度、固粒和气流的速度比值以及Stokes数不同时二维混合层流动中扰动频率与空间增长率的关系曲线,给出了关于悬浮固粒对流场失稳特性影响的几个重要结论。     

Phase trajectory portrait of the vibro-impact forced dynamics of two heavy mass particles motions along rough circle

Forced vibro-impact dynamics of the two heavy mass particle motions, in vertical plane, along rough circle with Coulomb’s type friction and one, one side impact limiter is considered in combinations of applied analytical and numerical methods. System of two differential double equations, each for one of two heavy mass particle motions along same rough circle are composed with corresponding initial conditions as well as impact conditions. By use software package tools differential double equations are numerically integrated for obtaining phase portrait of phase trajectory branches for different mass particles initial kinetic states. By series of the phase trajectory branches for each mass particle motion between two impacts or between impact and alternation of the Coulomb’s friction force direction, two phase trajectory graphs of the system vibro-impact non-linear dynamics are composed. Different software tools are used as helping tools for calculate time moments of the series of the impacts between mass particles, as well as positions of the impacts, necessary for calculations of the impact velocities of the mass particles before and after impacts. Some comparison between forced and free vibro-impact dynamics of the two heavy mass particles in vertical plane, along rough circle with Coulomb’s type friction and one, one side impact limiter is done. Trigger of coupled one side singularities in phase portraits are identified.     

Numerical simulation of dense fluid-particle flows using computational fluid dynamics (CFD) and discrete element method (DEM)

Discrete Element Method (DEM) has been successfully coupled with Computational Fluid Dynamics (CFD) in the framework of OpenFOAM an open source CFD simulation code. In the current study, at first the model is validated with the simple test case of spherical particle comparing the results with the analytical solution. Then the simulation of a gaseous fluidized bed is considered. The coupled mass and momentum balance equations are used to calculate the flow behavior, particle fluidization and bubble formation. The dimensions of the simulation domain are similar to Link et al. (2005) but with different stiffness of particles. The higher velocity of gaseous fluid relative to particles entering through a jet causes the particles to fluidize. The particles behavior, fluidization, bubble formation and the velocity vectors of particles show a good agreement with the literature. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Particle transport in a bottom-feed separation vessel

A two-dimensional, axisymmetric numerical model of particle separation in a bottom-feed separation vessel is presented. The model includes six separate particle classes and assumes that the settling velocity of each particle class is sufficiently small when compared to the high inflow turbulence levels that the effect of the particles on turbulence can be neglected. Low particle settling velocities coupled with low particle volume fractions allows application of a drift-flux multi-phase model. The comparison between numerical results and measured plant data is in good agreement for overflow of all particle classes. Results of simulations show that bottom feeding results in a negatively buoyant, particle-laden jet being formed in the core of the vessel. The fraction of large particles that is carried out through the overflow is found to be critically dependent on the inlet velocity. The most effective way to reduce carry-over of large particles at the same time as maintaining through-put is to increase the diameter of the inlet feed pipe.     

Numerical Study on the Influence of Particle Inertia in Particle-Driven Gravity Currents

We present two-dimensional numerical simulations of particle-driven gravity currents in a lock-exchange configuration. The fluid is described in an Eulerian framework, whereas the particles are tracked in a Lagrangian manner. The study is restricted to dilute suspensions, allowing to neglect particle-particle interactions. The particle forces considered are buoyancy and the Stokes drag. We study the influence of particle inertia on the flow evolution by performing simulations with different Stokes numbers. We also consider the case where particle inertia is neglected. Generally, we observe significant changes in the form and structure of the gravity current with increasing particle Stokes numbers. Particularly, the formation of Kelvin-Helmholtz vortices is more and more suppressed. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical inaccuracies across the interface of a nested grid

This article addresses and discusses the inaccuracies in finite differencing across the interface of a nested grid. Explicit schemes for the advection and diffusion equations are analyzed on the fine and coarse grids and reformulated at the interface to guarantee that the evolving solution is unaffected by the abrupt change of the spatial grid resolution. The associated errors are expressed as a function of the wavelength of the initial field distribution and the ratio between the coarse and fine grid resolution. It is found that large-scale features of the coarse grid must supply energy to sustain the small-scale features of the fine grid. To not deplete the large-scale motion, a source of energy must be given at the interface in the form of a computational diffusive term with negative viscosity coefficient. On the other hand, not all the energy of the small-scale features of the fine grid has to be transferred to the large-scale motion, but some of it needs to be computationally dissipated at the interface. © 1994 John Wiley & Sons, Inc.     

Parallel Smoothed Aggregation Multilevel Schwarz Preconditioned Newton-Krylov Algorithms for Poisson-Boltzmann Problems

We study a multilevel Schwarz preconditioned Newton-Krylov algorithm to solve the Poisson-Boltzmann equation with applications in multi-particle colloidal simulation. The smoothed aggregation-type coarse mesh space is introduced in collaboration with the one-level Schwarz method as a composite preconditioner for accelerating the convergence of a Krylov subspace method for solving the Jacobian system at each Newton step. The important feature of the proposed solution algorithm is that the geometric mesh information needed for constructing the multilevel preconditioner is the same as the one-level Schwarz method on the fine mesh. Other components, such as the definition of the coarse mesh, all the mesh transfer operators, and the coarse mesh problem, are taken care of by the Trillinos/ML packages of the Sandia National Laboratories in the United States. After algorithmic parameter tuning, we show that the proposed smoothed aggregation multilevel Newton-Krylov-Schwarz (NKS) algorithm numerically outperforms than smoothed aggregation multigrid method and one-level version of the NKS algorithm with satisfactory parallel performances up to a few thousand cores. Besides, we investigate how the electrostatic forces between particles for the separation distance depend on the radius of spherical colloidal particles and valence ratios of cation and anion in a cubic system.     

Computational fluid dynamics simulation of fluid particle fragmentation in turbulent flows

A simulation methodology is presented that allows detailed studies of the breakup mechanism of fluid particles in turbulent flows. The simulations, based on large eddy and volume of fluid simulations, agree very well with high-speed measurements of the breakup dynamics with respect to deformation time and length scales, and also the resulting size of the daughter fragments. The simulations reveal the size of the turbulent vortices that contribute to the breakup and how fast the interaction and energy transfer occurs. It is concluded that the axis of the deformed particle and the vortex core axis are aligned perpendicular to each other, and that breakup sometimes occurs due to interaction with two vortices at the same time. Analysis of the energy transfer from the continuous phase turbulence to the fluid particles reveals that the deformed particle attains it maximum in interfacial energy before the breakup is finalized. Similar to transition state theory in chemistry this implies that an activation barrier exists. Consequently, by considering the dynamics of the phenomenon, more energy than required at the final stage needs to be transferred from the turbulent vortices for breakup to occur. This knowledge helps developing new, more physical sound models for the breakup phenomenon required to solve scale separation problems in computational fluid dynamics simulations.     

Lagrangian coherent structures,transport and chaotic mixing in simple kinematic ocean models

Methods of dynamical system’s theory are used for numerical study of transport and mixing of passive particles (water masses, temperature, salinity, pollutants, etc.) in simple kinematic ocean models composed with the main Eulerian coherent structures in a randomly fluctuating ocean—a jet-like current and an eddy. Advection of passive tracers in a periodically-driven flow consisting of a background stream and an eddy (the model inspired by the phenomenon of topographic eddies over mountains in the ocean and atmosphere) is analyzed as an example of chaotic particle’s scattering and transport. A numerical analysis reveals a non-attracting chaotic invariant set Λ that determines scattering and trapping of particles from the incoming flow. It is shown that both the trapping time for particles in the mixing region and the number of times their trajectories wind around the vortex have hierarchical fractal structure as functions of the initial particle’s coordinates. Scattering functions are singular on a Cantor set of initial conditions, and this property should manifest itself by strong fluctuations of quantities measured in experiments. The Lagrangian structures in our numerical experiments are shown to be similar to those found in a recent laboratory dye experiment at Woods Hole. Transport and mixing of passive particles is studied in the kinematic model inspired by the interaction of a current (like the Gulf Stream or the Kuroshio) with an eddy in a noisy environment. We demonstrate a non-trivial phenomenon of noise-induced clustering of passive particles and propose a method to find such clusters in numerical experiments. These clusters are patches of advected particles which can move together in a random velocity field for comparatively long time. The clusters appear due to existence of regions of stability in the phase space which is the physical space in the advection problem.     

柱状固粒两相边界层流场稳定性研究

从运动方程和本构方程出发,推导得到了含柱状粒子两相流场的修正Orr-Sommerfeld方程,然后在边界层流场中,采用数值计算方法,得到了含柱状粒子流场的稳定性中性曲线,给出了流场失稳的临界雷诺数.结果表明在所述情况下,柱状粒子对流场起着抑制失稳的作用,而且抑制的程度随着柱状粒子体积分数和长径比的增加而提高.     

Dynamic model for a magnetorheological damper

A lumped mass thermo-mechanical model for the dynamics of a damper filled with a magnetorheological fluid is described, analyzed, and numerically simulated. The model includes friction and temperature effects, and consists of a differential inclusion for the piston displacements coupled with the energy balance equation for the temperature. The fluid viscosity is assumed to be a function the temperature and electrical current, which in practice may be used as the control variable. Numerical simulations of the system behavior are presented. In particular, the simulations of an initial impact show how the subsequent oscillations can be effectively damped.     

A non-isentropic Euler–Poisson model for a collisionless plasma

We analyse transonic solutions of the one-dimensional Euler–Poisson model for a collisionless gas of charged particles in the non-isentropic steady-state case. The model consists of the conservation of mass, momentum and energy equations. The electric field is modelled self-consistently (Coulomb field). Boundary conditions on the particle density and particle temperature are imposed. The analysis is based on representing solutions piecewise as orbits in the particle-density-electric-field phase plane and connecting the orbit segments by the jump and entropy conditions. We characterize the set of all solutions of the Euler–Poisson problem. In particular, we show that, depending upon the length of the interval on which the boundary value problem is posed, fully subsonic, one-shock and (in certain cases) two-shock transonic and smooth transonic solutions exist. Also, numerical computations illustrating the structure of the solutions are reported.