Macroscopic Conservation Equations Research Articles

MODELO MATEMÁTICO DETERMINÍSTICO-ESTOCÁSTICO PARA PREVISÃO DA MACROESTRUTURA BRUTA DE SOLIDIFICAÇÃO SOB EFEITO DA DECANTAÇÃO DE GRÃOS

A mathematical model is proposed to predict the as-cast macrostructure formation under the effect of grain settling. This model combines the macroscopic conservation equations for mass, energy, and chemical species with the cellular automaton technique to simulate the nucleation, growth, and settling of grains within the liquid metal. The model is applied to the solidification of a single grain settling in an undercooled liquid and showed an increase in temperature and solute concentration in the lower regions of the domain owing to the latent heat release and solute transfer from the grain envelopes to the liquid. The model is also applied to the upwards or downwards unidirectional solidification. In the upwards solidification, the settling of grains results in a completely columnar grain structure which otherwise, in the absence of settling, gives a mixed columnar and equiaxed grain structure. In the downwards solidification, the columnar grains growing downwards are blocked by a layer of equiaxed grains that settle and grow upwards at the ingot base.

Modelling feeding flow related shrinkage defects in aluminum castings

The process modelling of shape casting is geometrically complex and computationally very challenging. Besides the three-dimensional complex shapes with multiple domains, the defects of interest to industry arise as a consequence of the interaction amongst a range of phenomena. Conventionally, the key phenomena and defect prediction are modelled through empirical relations applied to the simulation results. Such approaches are neither comprehensive nor reliable. This paper presents a 3-D model that is capable of predicting the formation of shrinkage defects explicitly as a function of the interacting continuum phenomena, i.e. free surface flow, heat transfer, and solidification, in complex three-dimensional geometries which allows to identify the distinction between surface depression, surface connected cavities and internal cavities.The model solves the coupled macroscopic conservation equations for mass, momentum, and energy with a phase change during solidification. In the model, the volume deficit due to solidification can either be compensated by depression of the outside surface or by creating a cavity that initiates either on the surface or in the interior of the casting. The solidification morphology is taken into account by using a parameter, which depends on the fraction solid, in the momentum equation. By using an adapted free surface algorithm, it is suitable to predict surface connected defects: depressed surfaces and caved surfaces. A critical pressure serves as a criterion to open internal shrinkage cavities. The model does not need to search for connected zones to feed shrinkage, but the shrinkage distribution will automatically emerge from the continuity equation.This advanced shrinkage model has experimentally been validated successfully using two Al–Si alloys, a skin freezing eutectic alloy and a mushy freezing hypo-eutectic alloy.

Lattice Boltzmann method for one-dimensional radiation transfer

The macroscopic conservation equations of radiation energy and radiation momentum are derived on the basis of radiation hydrodynamics. Based on the Chapman-Enskog method, the lattice Boltzmann model for one-dimensional radiative transfer is proposed from the Boltzmann equation. The numerical simulation results agree well with the exact solution and show that the lattice Boltzmann method developed in this paper has good accuracy and stability for solving one-dimensional radiative transfer problems.

Investigation of Groundwater Contaminant Discharge into Tidally influenced Surface-water Bodies: Theoretical Analysis

Data from an one-dimensional homogeneous sand column, which is utilized to investigate the effect of tides on the concentration of groundwater contaminants discharging to a surface-water body, demonstrate that the tidal fluctuations in water level elevation create concentration oscillations upgradient of the groundwater discharge locations and there is a resulting decrease in average contaminant concentration at the point of groundwater discharge to a surface-water body. The further upgradient an observation point is located, the smaller the amplitude of the tidally induced concentration oscillations. In addition, an excessive upstream migration of concentration oscillations is observed although there is a net downgradient flow. As the classical groundwater flow and transport model could not reproduce this phenomena, a multi-mobility model is proposed with one highly mobile liquid phase, one less mobile liquid phase and a solid phase. Averaging theory is applied in a first step to develop the macroscopic mass conservation equation from its microscale counterpart and then, in a second step, averaging is again used to reduce dimensionality to one-dimensional governing equations defined along the axis of the column. The simulation confirms the existence of an enhanced tidally induced mixing process and the suitability of our mathematical-physical representation of it.

Modeling of Free Convection Heat Transfer to a Supercritical Fluid in a Square Enclosure by the Lattice Boltzmann Method

During the last decade, a number of numerical computations based on the finite volume approach have been reported, studying various aspects of heat transfer near the critical point. In this paper, a lattice Boltzmann method (LBM) has been developed to simulate laminar free convection heat transfer to a supercritical fluid in a square enclosure. The LBM is an ideal mesoscopic approach to solve nonlinear macroscopic conservation equations due to its simplicity and capability of parallelization. The lattice Boltzmann equation (LBE) represents the minimal form of the Boltzmann kinetic equation. The LBE is a very elegant and simple equation, for a discrete density distribution function, and is the basis of the LBM. For the mass and momentum equations, a LBM is used while the heat equation is solved numerically by a finite volume scheme. In this study, interparticle forces are taken into account for nonideal gases in order to simulate the velocity profile more accurately. The laminar free convection cavity flow has been extensively used as a benchmark test to evaluate the accuracy of the numerical code. It is found that the numerical results of this study are in good agreement with the experimental and numerical results reported in the literature. The results of the LBM-FVM (finite volume method) combination are found to be in excellent agreement with the FVM-FVM combination for the Navier–Stokes and heat transfer equations.

Interaction between single grain solidification and macrosegregation: Application of a cellular automaton—Finite element model

A two-dimensional (2D) cellular automaton (CA)—finite element (FE) model developed for the prediction of grain structure formation during solidification processes is presented. While the FE method solves the macroscopic conservation equations for heat and mass transfers, the CA method is used at a mesoscopic scale to simulate the growth of a mushy zone domain. The limit between this mushy zone domain and the undercooled liquid melt in which it develops defines the envelope of a single grain. A verification is first performed to check that the CAFE model retrieves the solution of a purely FE simulation for a vanishing undercooling of the growth front. Since the present 2D CAFE model is relevant with respect to the solidification experiment developed by Yin and Koster [H. Yin, J. N. Koster, J. Crystal Growth 205 (1999) 590; H. Yin, J. N. Koster, J. Alloy Compd. 352 (2003) 197], comparison is then performed. In particular, the position of the growth front is considered as a time sequence. The effect of the growth undercooling is clearly revealed. The kinetics of the growth front used in the model assumes the development of a rough interfacial dendrite-like morphology. This assumption is discussed considering the value of the normalized entropy of fusion of gallium.

A dual-scale coupled micro/macro segregation model for dendritic alloy solidification

The present article reports on the formulation, numerical implementation, and application of a single-domain coupled micro/macroscopic model for simulation of dendritic alloy solidification. Microscopic solutal non-equilibrium effects have been included in the macroscopic modeling of solidification by using a fixed grid dual scale numerical approach. Salient features of the present approach include a continuum model for conservation of mass, momentum, energy and species on the macroscopic scale, a microscopic solute redistribution model, and the solution procedure and auxiliary equations necessary for coupling between the two models. The coupling between macro and micro scale models is practically made possible by introducing an iterative micro/macro time step scheme. The local solidification rate is calculated by implicit iterations of macroscopic conservation equations and the microscopic solute redistribution model. The present model is capable to simulate eutectic reaction, local re-melting, and account for the inter-linkage between micro and macroscopic solute redistribution (micro and macrosegregation).

Non-local closure and parallel performance of lattice Boltzmann models for some plasma physics problems

The lattice Boltzmann (LB) method is a mesoscopic approach to solving nonlinear macroscopic conservation equations. Because the LB algorithm yields a simple collide-stream sequence it has been extensively applied to Navier–Stokes flows, but its MHD counterpart is less well known in the plasma physics community. Several plasma problems that should be amenable to LB are discussed. In particular, Landau damping—a collisionless kinetic phenomenon of wave–particle interaction—can be studied by LB since non-local macroscopic closures have been generated by plasma physicists. The parallel performance of 2D LB codes for MHD are presented, including scaling performance on the Earth Simulator.

Modelling convection in solidification processes using stabilized finite element techniques

AbstractSolidification of dendritic alloys is modelled using stabilized finite element techniques to study convection and macrosegregation driven by buoyancy and shrinkage. The adopted governing macroscopic conservation equations of momentum, energy and species transport are derived from their microscopic counterparts using the volume‐averaging method. A single domain model is considered with a fixed numerical grid and without boundary conditions applied explicitly on the freezing front. The mushy zone is modelled here as a porous medium with either an isotropic or an anisotropic permeability. The stabilized finite‐element scheme, previously developed by authors for modelling flows with phase change, is extended here to include effects of shrinkage, density changes and anisotropic permeability during solidification. The fluid flow scheme developed includes streamline‐upwind/Petrov–Galerkin (SUPG), pressure stabilizing/Petrov–Galerkin, Darcy stabilizing/Petrov–Galerkin and other stabilizing terms arising from changes in density in the mushy zone. For the energy and species equations a classical SUPG‐based finite element method is employed with minor modifications. The developed algorithms are first tested for a reference problem involving solidification of lead–tin alloy where the mushy zone is characterized by an isotropic permeability. Convergence studies are performed to validate the simulation results. Solidification of the same alloy in the absence of shrinkage is studied to observe differences in macrosegregation. Vertical solidification of a lead–tin alloy, where the mushy zone is characterized by an anisotropic permeability, is then simulated. The main aim here is to study convection and demonstrate formation of freckles and channels due to macrosegregation. The ability of stabilized finite element methods to model a wide variety of solidification problems with varying underlying phenomena in two and three dimensions is demonstrated through these examples. Copyright © 2005 John Wiley & Sons, Ltd.

Hierarchical Approach to Model Multilayer Colloidal Deposition in Porous Media

Particle deposition is important in many environmental systems such as water and wastewater filtration, air pollution control, subsurface transport, biofilm formation and fouling, and thin film synthesis for use in remediation technologies. While continuum-level models have been developed to predict deposition dynamics in these systems, these models fail to explain transient dynamics of multilayer deposition from a mechanistic viewpoint. In this work, a multiscale approach has been developed to predict multiple layer irreversible colloidal deposition in the presence of interparticle electrostatic and van der Waals interactions in porous media. The approach combines the kinetic information obtained from the mesoscopic stochastic simulations of particle deposition with the macroscopic conservation equation describing colloidal transport. Sequential Brownian dynamics simulations are first performed by accounting for particle-particle (P-P) and particle-surface (P-S) interactions, and multilayered particle deposits are obtained. The available surface function quantifying the deposition kinetics is then obtained from the deposit microstructure. Deposition dynamics are studied at different ionic strengths and particle potentials that control the range and magnitude of interparticle interactions. Simulation results showed that the microstructure of the particle deposits formed under the influence of P-P and P-S electrostatic interactions exhibited significant variations with respect to ionic strength and could be qualitatively explained bythe interplay between the repulsive and attractive P-P and P-S interaction forces. The available surface function also varied significantly as a function of ionic strength. This basic understanding of the deposition dynamics at the mesoscale was then combined with the continuum-level transport equations to predict particle breakthrough curves in porous media. The approach is capable of capturing transient features of deposition dynamics, as demonstrated by the good agreement between the model predictions and the experimental observations.

Modelling of free boundary problems for phase change with diffuseinterfaces

We present a continuum thermodynamical framework for simulating multiphase Stefan problem. For alloy solidification, which is marked by a diffuse interface called the mushy zone, we present a phase filed like formalism which comprises a set of macroscopic conservation equations with an order parameter which can account for the solid, liquid, and the mushy zones with the help of a phase function defined on the basis of the liquid fraction, the Gibbs relation, and the phase diagram with local approximations. Using the above formalism for alloy solidification, the width of the diffuse interface (mushy zone) was computed rather accurately for iron‐carbon and ammonium chloride‐water binary alloys and validated against experimental data from literature.

Comparison between theoretical values and simulation results of viscosity for the dissipative particle dynamics method

In order to investigate the validity of the dissipative particle dynamics method, which is a mesoscopic simulation technique, we have derived an expression for viscosity from the equation of motion of dissipative particles. In the concrete, we have shown the Fokker–Planck equation in phase space, and macroscopic conservation equations such as the equation of continuity and the equation of momentum conservation. The basic equations of the single-particle and pair distribution functions have been derived using the Fokker–Planck equation. The solutions of these distribution functions have approximately been solved by the perturbation method under the assumption of molecular chaos. The expressions of the viscosity due to momentum and dissipative forces have been obtained using the approximate solutions of the distribution functions. Also, we have conducted nonequilibrium dynamics simulations to investigate the influence of the parameters, which have appeared in defining the equation of motion in the dissipative particle dynamics method. The theoretical values of the viscosity due to dissipative forces in the Hoogerbrugge–Koelman theory are in good agreement with the simulation results obtained by the nonequilibrium dynamics method, except in the range of small number densities. There are restriction conditions for taking appropriate values of the number density, number of particles, time interval, shear rate, etc., to obtain physically reasonable results by means of dissipative particle dynamics simulations.

Transport properties of partially ionized and unmagnetized plasmas

This work is a comprehensive and theoretical study of transport phenomena in partially ionized and unmagnetized plasmas by means of kinetic theory. The pros and cons of different models encountered in the literature are presented. A dimensional analysis of the Boltzmann equation deals with the disparity of mass between electrons and heavy particles and yields the epochal relaxation concept. First, electrons and heavy particles exhibit distinct kinetic time scales and may have different translational temperatures. The hydrodynamic velocity is assumed to be identical for both types of species. Second, at the hydrodynamic time scale the energy exchanged between electrons and heavy particles tends to equalize both temperatures. Global and species macroscopic fluid conservation equations are given. New constrained integral equations are derived from a modified Chapman-Enskog perturbative method. Adequate bracket integrals are introduced to treat thermal nonequilibrium. A symmetric mathematical formalism is preferred for physical and numerical standpoints. A Laguerre-Sonine polynomial expansion allows for systems of transport to be derived. Momentum, mass, and energy fluxes are associated to shear viscosity, diffusion coefficients, thermal diffusion coefficients, and thermal conductivities. A Goldstein expansion of the perturbation function provides explicit expressions of the thermal diffusion ratios and measurable thermal conductivities. Thermal diffusion terms already found in the Russian literature ensure the exact mass conservation. A generalized Stefan-Maxwell equation is derived following the method of Kolesnikov and Tirskiy. The bracket integral reduction in terms of transport collision integrals is presented in Appendix for the thermal nonequilibrium case. A simple Eucken correction is proposed to deal with the internal degrees of freedom of atoms and polyatomic molecules, neglecting inelastic collisions. The authors believe that the final expressions are readily usable for practical applications in fluid dynamics.

Multi-level lattice Boltzmann model on square lattice for compressible flows

A compressible lattice Boltzmann model is established on a square lattice. The model allows large variations in the mean velocity by introducing a large particle-velocity set. To maintain tractability, the support set of the equilibrium distribution is chosen to include only four directions and three particle-velocity levels in which the third level is introduced to improve the stability of the model. This simple structure of the equilibrium distribution makes the model efficient for the simulation of flows over a wide range of Mach numbers and gives it the capability of capturing shock jumps. Unlike the standard lattice Boltzmann model, the formulation eliminated the fourth-order velocity tensors, which were the source of concerns over the homogeneity of square lattices. A modified collision invariant eliminates the second-order discretization error of the fluctuation velocity in the macroscopic conservation equation from which the Navier–Stokes equation and energy equation are recovered. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Two-dimensional shock-wave propagations and boundary layer flows were successfully simulated. The model can be easily extended to three-dimensional cubic lattices.

A stabilized volume‐averaging finite element method for flow in porous media and binary alloy solidification processes

AbstractA stabilized equal‐order velocity–pressure finite element algorithm is presented for the analysis of flow in porous media and in the solidification of binary alloys. The adopted governing macroscopic conservation equations of momentum, energy and species transport are derived from their microscopic counterparts using the volume‐averaging method. The analysis is performed in a single domain with a fixed numerical grid. The fluid flow scheme developed includes SUPG (streamline‐upwind/Petrov–Galerkin), PSPG (pressure stabilizing/Petrov–Galerkin) and DSPG (Darcy stabilizing/Petrov–Galerkin) stabilization terms in a variable porosity medium. For the energy and species equations a classical SUPG‐based finite element method is employed. The developed algorithms were tested extensively with bilinear elements and were shown to perform stably and with nearly quadratic convergence in high Rayleigh number flows in varying porosity media. Examples are shown in natural and double diffusive convection in porous media and in the directional solidification of a binary‐alloy. Copyright © 2004 John Wiley & Sons, Ltd.

Comparison between Theoretical Values and Simulation Results of Transport Coefficients for the Dissipative Particle Dynamics Method (1st Report, Relationship between the Equation of Motion of Dissipative Particles and the Theoretical Solution of Transport Coefficients)

For the first step to clarify the validity of the dissipative particle dynamics method, which is a mesoscopic simulation technique, we have derived the expression for the transport coefficient such as the viscosity, from the equation of motion of the dissipative particles. In the concrete, we have first shown the Fokker-Planck equation in phae space, and the macroscopic conservation equations such as the equation of continuity and the equation of momentum conservation. Then, we have derived the basic equations of the single-particle and pair distribution functions using the Fokker-Planck equation. The solutions of these distribution functions have approximately been solved by the mehod of perturbation method under the assumption of the molecular chaos.Finally, the expression of the viscosity due to the momentum and dissipative forces has been obtained using the approximate solutions of the distributions.

Three-dimensional lattice Boltzmann model for compressible flows.

A three-dimensional compressible lattice Boltzmann model is formulated on a cubic lattice. A very large particle-velocity set is incorporated in order to enable a greater variation in the mean velocity. Meanwhile, the support set of the equilibrium distribution has only six directions. Therefore, this model can efficiently handle flows over a wide range of Mach numbers and capture shock waves. Due to the simple form of the equilibrium distribution, the fourth-order velocity tensors are not involved in the formulation. Unlike the standard lattice Boltzmann model, no special treatment is required for the homogeneity of fourth-order velocity tensors on square lattices. The Navier-Stokes equations were recovered, using the Chapman-Enskog method from the Bhatnagar-Gross-Krook (BGK) lattice Boltzmann equation. The second-order discretization error of the fluctuation velocity in the macroscopic conservation equation was eliminated by means of a modified collision invariant. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Since the present scheme deals only with the equilibrium distribution that depends only on fluid density, velocity, and internal energy, boundary conditions on curved wall are easily implemented by an extrapolation of macroscopic variables. To verify the scheme for inviscid flows, we have successfully simulated a three-dimensional shock-wave propagation in a box and a normal shock of Mach number 10 over a wedge. As an application to viscous flows, we have simulated a flat plate boundary layer flow, flow over a cylinder, and a transonic flow over a NACA0012 airfoil cascade.

Averaged solute transport during solidification of a binary mixture: Active dispersion in dendritic structures

The macroscopic conservation equations governing solute transport during solidification of binary alloys are derived using an averaging procedure in the context of macroscale nonequilibrium. Special attention is focused on the derivation of the associated closure problems, leading to the determination of the effective dispersion tensor and macroscopic interphase coefficients that characterize active dispersion phenomena. These closure problems are solved numerically using schematic structures and digitized images of real columnar dendritic structures observed experimentally during solidification of succinonitrile-4 wt pct acetone. The influence of the geometry and dispersion on the effective solute-properties transport is analyzed, and comparison with passive dispersion is provided. The theoretical and numerical results indicate, first, that tortuosity effects are small, and this is related to the impact of the boundary condition at the interface between the two phases, and, second, that dispersion becomes very important only at very large Peclet numbers.

Passive dispersion in dendritic structures

In order to improve mass transport description in solidification modeling, this study deals with the determination of the solute dispersion tensor in columnar dendritic mushy zone. The closure problem associated with the derivation of the macroscopic species conservation equation, using the volume averaging method, is solved numerically. Using schematic structures and digitized images of real dendritic structures observed experimentally during solidification of succinonitrile-4 wt.% acetone, the influence of tortuosity and microscopic dispersion on the determination of the effective solute dispersion coefficients is represented in terms of the Péclet number.

Lattice-Boltzmann models for high speed flows

We formulate a lattice Boltzmann model to solve supersonic flows. The particle velocities are determined by the mean velocity and internal energy. The adaptive nature of particle velocities permits the mean flow to have a high Mach number. The introduction of a particle potential energy causes the model to be suitable for a perfect gas with an arbitrary specific heat ratio. The macroscopic conservation equations are derived by the Chapman-Enskog method. The simulations were carried out on the hexagonal lattice. However, the extension to both two- and three-dimensional square lattices is straightforward. As preliminary tests, we present the Sod shock-tube simulation and the two-dimensional shock reflection simulation.