Expansion into vacuum of a gaseous ellipsoid with repulsion between the particles

Methods of the analytic theory of differential equations as well as the numerical methods are used to study the problem of adiabatic expansion of a homogeneous cloud of perfect gas into vacuum. The Cartesian and Lagrangian coordinates of the gas particles are connected by a linear relation, and the particles have a charge which generates a macroscopic field within the cloud, without however affecting the properties of the perfect gas.     

Eulerian and Lagrangian predictions of particulate two-phase flows: a numerical study

To predict particulate two-phase flows, two approaches are possible. One treats the fluid phase as a continuum and the particulate second phase as single particles. This approach, which predicts the particle trajectories in the fluid phase as a result of forces acting on particles, is called the Lagrangian approach. Treating the solid as some kind of continuum, and solving the appropriate continuum equations for the fluid and particle phases, is referred to as the Eulerian approach.Both approaches are discussed and their basic equations for the particle and fluid phases as well as their numerical treatment are presented. Particular attention is given to the interactions between both phases and their mathematical formulations. The resulting computer codes are discussed.The following cases are presented in detail: vertical pipe flow with various particle concentrations; and sudden expansion in a vertical pipe flow. The results show good agreement between both types of approach.The Lagrangian approach has some advantages for predicting those particulate flows in which large particle accelerations occur. It can also handle particulate two-phase flows consisting of polydispersed particle size distributions. The Eulerian approach seems to have advantages in all flow cases where high particle concentrations occur and where the high void fraction of the flow becomes a dominating flow controlling parameter.     

Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model

In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacts with all others. Near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community. In this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full-interaction problem so as to answer the question mathematically for a one-dimensional model. A few numerical methods have been designed in the engineering literature for the simplified model. Recently much attention has been paid to a finite-element-like quasicontinuum (QC) method which utilizes a mixed atomistic/continuum approximation model. No numerical analysis has been done yet. In the paper we shall estimate the error of the QC method for this one-dimensional model. Possible ill-posedness of the method and its modification are discussed as well.

     

Transient conduction in a plate with counteracting convection and thermal radiation at the boundaries

During the flash dehydroxylation of powdered kaolinite it is desirable that a rapidly propagating thermal wave penetrates the cold powder particles in a way that raises the particle interior to the reaction temperature of 600°C without the particle exterior being heated beyond 1000°C. In a production unit this is achieved by performing the heat treatment in a device where particles are heated by convection from hot gas and are subject to heat loss by thermal radiation to cool walls. This paper concerns the fundamental heat transfer problem of the process, decoupled from the thermal effects of the dehydroxylation reaction. Using a plate as the approximation for the particle shape a semi-analytical solution for the plate temperature distribution is obtained as a function of the five dimensionless process parameters: Biot number, radiation number, wall/gas and particle/gas temperature ratios and mode of convection. Accuracy is demonstrated by comparison with an existing numerical solution for the limiting case of pure radiative heating of a plate initially at absolute zero.     

3D gradient corrected SPH for fully resolved particle–fluid interactions

Fully resolved fluid–solid coupling is explored with the gradient corrected weakly compressible SPH methodology being used to simulate an incompressible Newtonian fluid as well as being used to obtain the coupling force information required to accurately represent these interactions. Gradient correction allows for the application of the Neumann boundary condition required to describe the pressure fields at solid interfaces, as well as symmetry boundary conditions for velocity (where applicable) without the use of ghost or mirrored particles. A scaling study is performed by investigating the drag on an infinitely long cylinder at different smoothed particle hydrodynamics (SPH) resolutions, with finer resolution scales showing good correlation to other studies. The drag characteristics of several particle shapes and topologies are also investigated making use of both convex and non-convex particle shapes. Clear distinction for both the fluid and solid particle responses for the various solid particle shapes are observed. Boundary effects are also explored with results showing a strong responses to changing domain geometry aspect ratios. A many particle system with two different particle shapes are simulated to investigate bulk behaviour of the different solids falling under gravity in a fluid. All results presented in this paper are obtained from full 3D simulations.     

Numerical simulation of the performance of a snow fence with airfoil snow plates by FVM

The aim of this work is to analyze the efficiency of a snow fence with airfoil snow plates to avoid the snowdrift formation, to improve visibility and to prevent blowing snow disasters on highways and railways. In order to attain this objective, it is necessary to solve particle transport equations along with the turbulent fluid flow equations since there are two phases: solid phase (snow particles) and fluid phase (air). In the first place, the turbulent flow is modelled by solving the Reynolds-averaged Navier-Stokes (RANS) equations for incompressible viscous flows through the finite volume method (FVM) and then, once the flow velocity field has been determined, representative particles are tracked using the Lagrangian approach. Within the particle transport models, we have used a particle transport model termed as Lagrangian particle tracking model, where particulates are tracked through the flow in a Lagrangian way. The full particulate phase is modelled by just a sample of about 15,000 individual particles. The tracking is carried out by forming a set of ordinary differential equations in time for each particle, consisting of equations for position and velocity. These equations are then integrated using a simple integration method to calculate the behaviour of the particles as they traverse the flow domain. Finally, the conclusions of this work are exposed.     

Numerical Simulation of Detonation Excitation at Shock Wave Interaction with Dust Cloud in Channel

The study of detonation ability of reactive particle gas mixtures is necessary to prevent industrial explosions in industries where dispersed powders are used. The present paper focuses on numerical simulation of the shock wave interaction with semiinfinite aluminum dust cloud, which is situated inside a plane channel. The cloud fills entirely or partly the channel cross‐section and has initially a rectangular shape. The scenarios of detonation initiation in the cloud are determined depending on the incident shock wave amplitude values. The processes of transformation and spreading of finite width clouds under weak incident shock wave action (when the particles do not ignite) are investigated. The types of an oblique shock wave reflection from the plane of symmetry in the cloud are analyzed. The processes of particle ignition and detonation structure formation at strong incident shock wave action are investigated. Nonstationary periodic fuctuations take place in the detonation flow due to transversal wave effect. Nevertheless the detonation structure established propagates in quasistationary regime. If the incident shock wave is attenuated with a rarefaction wave then the detonation formation fails at clouds of insufficient width.     

CFD of turbulent transport of particles behind a backward-facing step using a new model—k-ε-Sp

The k-ε-S_p model, describing two-dimensional gas–solid two-phase turbulent flow, has been developed. In this model, the diffusion flux and slip velocity of solid particles are introduced to represent the particle motion in two-phase flow. Based on this model, the gas–solid two-phase turbulent flow behind a vertical backward-facing step is simulated numerically and the turbulent transport velocities of solid particles with high density behind the step are predicted. The numerical simulation is validated by comparing the results of the numerical calculation with two other two-phase turbulent flow models (k-ε-A_p, k-ε-k_p) by Laslandes and the experimental measurements. This model, not only has the same virtues of predicting the longitudinal transport of the solid particles as the present practical two-phase flow models, but also can predict the lateral transport of the solid particles correctly.     

Ideal gas approximation for a two-dimensional rarefied gas under Kawasaki dynamics

In this paper we consider a two-dimensional lattice gas under Kawasaki dynamics, i.e., particles hop around randomly subject to hard-core repulsion and nearest-neighbor attraction. We show that, at fixed temperature and in the limit as the particle density tends to zero, such a gas evolves in a way that is close to an ideal gas, where particles have no interaction. In particular, we prove three theorems showing that particle trajectories are non-superdiffusive and have a diffusive spread-out property. We also consider the situation where the temperature and the particle density tend to zero simultaneously and focus on three regimes corresponding to the stable, the metastable and the unstable gas, respectively.     

Steady state numerical simulation of the particle collection efficiency of a new urban sustainable gravity settler using design of experiments by FVM

The aim of this work is to analyze the efficiency of a new sustainable urban gravity settler to avoid the solid particle transport, to improve the water waste quality and to prevent pollution problems due to rain water harvesting in areas with no drainage pavement. In order to get this objective, it is necessary to solve particle transport equations along with the turbulent fluid flow equations since there are two phases: solid phase (sand particles) and fluid phase (water). In the first place, the turbulent flow is modelled by solving the Reynolds-averaged Navier-Stokes (RANS) equations for incompressible viscous flows through the finite volume method (FVM) and then, once the flow velocity field has been determined, representative particles are tracked using the Lagrangian approach. Within the particle transport models, a particle transport model termed as Lagrangian particle tracking model is used, where particulates are tracked through the flow in a Lagrangian way. The full particulate phase is modelled by just a sample of about 2,000 individual particles. The tracking is carried out by forming a set of ordinary differential equations in time for each particle, consisting of equations for position and velocity. These equations are then integrated using a simple integration method to calculate the behaviour of the particles as they traverse the flow domain. The entire FVM model is built and the design of experiments (DOE) method was used to limit the number of simulations required, saving on the computational time significantly needed to arrive at the optimum configuration of the settler. Finally, conclusions of this work are exposed.     

A unit commitment-based fuzzy bilevel electricity trading model under load uncertainty

In this study, we establish a bilevel electricity trading model where fuzzy set theory is applied to address future load uncertainty, system reliability as well as human imprecise knowledge. From the literature, there have been some studies focused on this bilevel problem while few of them consider future load uncertainty and unit commitment optimization which handles the collaboration of generation units. Then, our study makes the following contributions: First, the future load uncertainty is characterized by fuzzy set theory, as the various factors that affect the load forecasting are often assessed with some non-statistical uncertainties. Second, the generation costs are obtained by solving complicated unit commitment problems, rather than approximate calculations used in existing studies. Third, this model copes with the optimizations of both the generation companies and the market operator, where the unexpected load risk is particularly analyzed by using fuzzy value-at-risk as a quantitative risk measurement. Forth, a mechanism to encourage the convergence of the bilevel model is proposed based on fuzzy maxmin approach, and a bilevel particle swarm optimization algorithm is developed to solve the problem in a proper runtime. To illustrate the effectiveness of this research, we provide a test system-based numerical example and discuss about the experimental results according to the principle of social welfare maximization. Finally, we also compare the model and algorithm with conventional methods.     

Mathematical modelling of bottom evolution by particle deposition and resuspension

A two-dimensional numerical model for the evolution of a bottom due to particle deposition and resuspension by a fluid flow is here presented. A computational fluid dynamic approach is used to calculate the flow field and a Lagrangian particle tracking technique is applied to solve the dispersed phase. The evolution of the lower boundary is simulated taking into account the mass conservation of the solid phase and the geotechnical properties of the granular material. The model is characterized by two important features. First, fluid dynamics are coupled with the bottom evolution due to particle deposition and resuspension. This permits to use the model to simulate complex flow fields as well as complex time-evolving geometries. Second, the dispersed phase is calculated by a Lagrangian approach, which retains the discrete information of the individual particles of the granular bottom which may be of interest for some industrial processes (coating) and environmental flows (sediment stratification). First consistency checks have been performed for some deposition and resuspension test cases with fluid at rest. The model has also been tested by comparison with a physical experiment of deposition inside a cavity. Finally, as an example of possible applications of industrial and environmental interest, the model has been applied to investigate particle deposition in rectangular cavities and the evolution of a sand heap by a fluid flow.     

Solving two-phase freezing Stefan problems: Stability and monotonicity

The two-phase Stefan problems with phase formation and depletion are special cases of moving boundary problems with interest in science and industry. In this work, we study a solidification problem, introducing a front-fixing transformation. The resulting non-linear partial differential system involves singularities, both at the beginning of the freezing process and when the depletion is complete, that are treated with special attention in the numerical modelling. The problem is decomposed in three stages, in which implicit and explicit finite difference schemes are used. Numerical analysis reveals qualitative properties of the numerical solution spatial monotonicity of both solid and liquid temperatures and the evolution of the solidification front. Numerical experiments illustrate the behaviour of the temperatures profiles with time, as well as the dynamics of the solidification front.     

Mathematical Analysis of a Coarsening Model with Local Interactions

We consider particles on a one-dimensional lattice whose evolution is governed by nearest-neighbor interactions where particles that have reached size zero are removed from the system. Concentrating on configurations with infinitely many particles, we prove existence of solutions under a reasonable density assumption on the initial data and show that the vanishing of particles and the localized interactions can lead to non-uniqueness. Moreover, we provide a rigorous upper coarsening estimate and discuss generic statistical properties as well as some non-generic behavior of the evolution by means of heuristic arguments and numerical observations.     

Dispersion of solid particles in turbulant flow through pipe expansions

Studies of the effects upon gas-liquid two-phase flows of pipefittings such as expansions, contractions, bends, and valveshave usually concentrated upon pressuredrop correlations andhave not attempted to determine changes in the distributionsof the gas and liquid phases caused by the fitting. However,it is known that such information is important if, for example,flow separators, which divide the gas and liquid phases in avariety of industrial processes, are to function efficiently.It is therefore important to gain an understanding of the influenceupon phase distributions of the common pipe fittings mentionedabove, which will be found in almost any industrial pipeworksystem. As a first step, the dispersion of solid particles carriedby turbulent gas flows through a pipe expansion has been modellednumerically. The commercial fluid-flow code CFDS-FLOW{smalltilde}hDas been used to model the gas flow, together with aneddy interaction model for determination of the motion of thesolid particles. Mean particle velocities and root-mean-squarevalues of the particle velocity fluctuations, as well as particleconcentrations, are evaluated and compared with recent experimentalresults. The influence of different eddy-length and eddy-lifetimespecifications upon the dispersion of particles of various sizesis investigated. It is found that the different eddy characteristicshave little effect on predicted mean particle velocities, whereasfluctuations in particle velocities and particle concentrationare sensitive to the changes made. By comparing the resultswith experimental data, it is possible to draw conclusions aboutthe relative merits of the different eddy specifications.     

Smoothed Particle ElectroMagnetics: A mesh-free solver for transients

In this paper an advanced mesh-free particle method for electromagnetic transient analysis, is presented. The aim is to obtain efficient simulations by avoiding the use of a mesh such as in the most popular grid-based numerical methods. The basic idea is to obtain numerical solutions for partial differential equations describing the electromagnetic problem by using a set of particles arbitrarily placed in the problem domain. The mesh-free smoothed particle hydrodynamics method has been adopted to obtain numerical solution of time domain Maxwell's curl equations. An explicit finite difference scheme has been employed for time integration. Details about the numerical treatment of electromagnetic vector fields components are discussed. Two case studies in one and in two dimensions are reported. In order to validate the new proposed methodology, named as Smoothed Particle ElectroMagnetics, a comparison with the standard finite difference time domain method results is performed. The intrinsic adaptive capability of the proposed method, has been exploited by introducing irregular particles distribution.     

Numerical simulation of pollution and oil spill spreading by the stochastic discrete particle method

The features of the stochastic discrete particle method are discussed as applied to the simulation of pollutant advection and diffusion in a turbulent flow and to the spread of a thin film of a viscous substance (oil) on the surface of water. The diffusion tensor in the former problem depends on the scale of the pollution cloud, and the diffusivity in the latter problem depends nonlinearly on the desired function. For pollution dispersion by a turbulent flow, a stochastic discrete particle algorithm is constructed in the case when the diffusion tensor corresponds to the Richardson 4/3 law. The numerical and analytical results are shown to agree well. The problem of oil film spreading is described by a quasilinear advection-diffusion equation. For this problem, a random walking algorithm is constructed in which the variance of the walking particle step depends on the desired function. For both instantaneous and time-continuous sources of pollutants, the solution produced by the stochastic discrete particle method agrees well with the analytical and/or numerical solutions to the test problems under consideration.     

Particle simulation of granular media and homogenisation towards continuum quantities

From a microscopic point of view, various natural and engineering materials consist of individual grains, whose motion strongly influence the macroscopic material behaviour. Exemplarily, one may look at the development of shear zones in natural granular materials, such as sand, occurring as a result of local grain dislocations and the transition of the granulate from a denser to a looser packing. The intuitive modelling approach for granular assemblies is consequently the consideration of each grain as a rigid particle. In a numerical framework, this leads to the Discrete Element Method (DEM), wherein the motion of each particle can be obtained solving Newton's equations for each particle. The present contribution discusses the basic fundaments of modelling granular material on the microscopic scale by use of the DEM. Special interest is taken to the constitutive choice of the governing particle-to-particle contact forces, as they have to account for plastic material behaviour as well as for assumptions concerning particle shape, size and distribution. As engineering problems are regularly described on the macroscale by means of continuum mechanics, a homogenisation strategy transfers the information from the microscale towards continuum quantities via volume averaging. Therefore, characteristic Representative Elementary Volumes (REV) are constructed by an ensemble of particles, where each particle can be chosen as the centre of a REV. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical investigation of blood flow. Part I: In microvessel bifurcations

In some diseases there is a focal pattern of velocity in regions of bifurcation, and thus the dynamics of bifurcation has been investigated in this work. A computational model of blood flow through branching geometries has been used to investigate the influence of bifurcation on blood flow distribution. The flow analysis applies the time-dependent, three-dimensional, incompressible Navier–Stokes equations for Newtonian fluids. The governing equations of mass and momentum conservation were solved to calculate the pressure and velocity fields. Movement of blood flow from an arteriole to a venule via a capillary has been simulated using the volume of fluid (VOF) method. The proposed simulation method would be a useful tool in understanding the hydrodynamics of blood flow where the interaction between the RBC deformation and blood flow movement is important. Discrete particle simulation has been used to simulate the blood flow in a bifurcation with solid and fluid particles. The fluid particle method allows for modeling the plasma as a particle ensemble, where each particle represents a collective unit of fluid, which is defined by its mass, moment of inertia, and translational and angular momenta. These kinds of simulations open a new way for modeling the dynamics of complex, viscoelastic fluids at the micro-scale, where both liquid and solid phases are treated with discrete particles.     

Randomized and quantum algorithms yield a speed-up for initial-value problems

Quantum algorithms and complexity have recently been studied not only for discrete, but also for some numerical problems. Most attention has been paid so far to the integration and approximation problems, for which a speed-up is shown in many important cases by quantum computers with respect to deterministic and randomized algorithms on a classical computer. In this paper, we deal with the randomized and quantum complexity of initial-value problems. For this nonlinear problem, we show that both randomized and quantum algorithms yield a speed-up over deterministic algorithms. Upper bounds on the complexity in the randomized and quantum settings are shown by constructing algorithms with a suitable cost, where the construction is based on integral information. Lower bounds result from the respective bounds for the integration problem.