Explicit Finite Difference Solution Research Articles

Phase Field Modeling of Air Entrapment in Binary Droplet Impact with Solidification Microstructure Formation

A novel numerical model was developed to investigate air entrapment in binary droplet impact with solidification microstructure formation under practical plasma spraying conditions. The evolving liquid–gas interface was tracked by the explicit finite difference solution to the Cahn–Hilliard equation, coupled with the Navier–Stokes equations. Another diffuse interface model was invoked to trace solid–liquid and grain–grain boundaries. The model was discretized using an explicit finite difference method on a half-staggered grid. The velocity pressure coupling was decoupled with the projection method. The in-house code was written in Fortran and was run with the aid of the shared memory parallelism, OpenMP. The time duration over which gas compressibility matters was estimated. Typical cases with air entrapment were studied with the model. The effect of droplet porosity on air entrapment was probed into as well: the larger the porosity of a droplet, the bigger the trapped air bubble. The grain growth near the air bubble is skewed. Moreover, a case without air entrapment was also shown herein to stress that air bubbles could be suppressed or even eliminated in plasma spraying by adjusting the landing positions of successive droplets.

Study on Heat Transfer Characteristics and Intelligent Optimization of Protective Clothing Materials in front of the Blast Furnace.

Special protective clothing should have a certain thermal insulation and protective effect in the production in front of the blast furnace. In order to study the design of special protective clothing for blast furnace foreman, its temperature distribution and internal thickness structure are deeply analyzed. First, based on the existing experimental data, combined with heat transfer and the Fourier law theory, a one-dimensional unsteady heat conduction equation is established. Using the explicit finite difference solution algorithm, the distribution law of the temperature of each fabric layer with time is obtained and the error analysis is performed. The error result is 0.42877%, which shows that the numerical solution is stable and convergent. Then, through the intelligent optimization model of a simulated annealing ant colony (SA-ACO), compared with the single ant colony algorithm (ACO) and a simulated annealing algorithm (SA), it avoids falling into the local optimal solution, reversely solves the optimal thickness of two layers of protective clothing in front of the blast furnace, improves the convergence speed of optimization, and verifies the reliability of the model with a sensitivity test. Finally, this paper actually solves the problem of the protective clothing design and mathematical intelligent model in front of the blast furnace and combines it with numerical simulation software to realize the numerical simulation of temperature distribution and thickness optimization of special protective clothing in front of the blast furnace, so as to provide theoretical guidance for enterprise blast furnace production and development of special protective clothing.

Numerical investigation of solidification microstructure formation in sequential YSZ droplet impact under supersonic plasma spraying

A coupled CFD and diffuse interface model was developed to predict the dynamic process of solidification microstructure formation of sequential yttria-stabilized zirconia (YSZ) droplet impact under supersonic plasma spraying. The numerical model relies on the explicit finite difference solution of the Navier-Stokes and energy balance equations, coupled with the Cahn-Hilliard equation, to track liquid-gas interface, and of a phase field model for solidification microstructure formation involving polycrystalline growth to trace solid-liquid interface. Extensive simulations, differing in the status of the first droplet and the impacting parameters of the second one, were carried out. The results reveal that solidification microstructure still follows a columnar pattern if the second droplet lands on the first that is just starting spreading, and that the growth direction around the contact region in the second splat will be obviously skewed by fluid flow if the first has been solidified, owing to earlier and easier nucleation on the top surface of the first splat, thus confirming the epitaxial growth across the splat-splat interface. Computed results are also compared with thermal spray experiments, with gratifying agreement obtained.

Numerical modeling of YSZ droplet impact/spreading with solidification microstructure formation in plasma spraying

A comprehensive numerical model has been developed, which is capable of representing the impact and spreading of a high-velocity yttria-stabilized zirconia (YSZ) molten droplet with microstructure formation in plasma thermal spraying processes. The numerical model entails the explicit finite difference solution of the Navier-Stokes equations and the energy balance equations, coupled with the Cahn-Hilliard equation to track the liquid-gas two phase interface and with a phase field model for solidification microstructure formation involving the polycrystalline growth. Numerical model procedures are given. The model, after being checked for mesh-independence, was applied to study the spreading and solidification of a high-temperature high-velocity YSZ droplet impinging upon a preheated substrate surface under the conditions typical of supersonic plasma thermal spraying processes. Extensive numerical simulations were carried out and results reveal that the solidification microstructure formed in the spreading droplet consists primarily of columnar grains. Computed results are also compared with thermal spray experiments, and gratifying agreement is obtained.

Explicit finite difference solution for contaminant transport problems with constant and oscillating boundary conditions

For constant and oscillating boundary conditions, the 1-D advection-diffusion equation with constant coefficients, which describes a contaminant flow, is solved by the explicit finite difference method in a semi-infinite medium. It is shown how far the periodicity of the oscillating boundary carries on until diminishing to below appreciable levels a specified distance away, which depends on the oscillation characteristics of the source. Results are tested against an analytical solution reported for a special case. The explicit finite difference method is shown to be effective for solving the advection-diffusion equation with constant coefficients in semi-infinite media with constant and oscillating boundary conditions.

Explicit Finite Difference Solution for One Dimensional Solute Transport of Unstable Flow

Explicit Finite Difference Solution for One Dimensional Solute Transport of Unstable Flow

Explicit finite-difference solution of two-dimensional solute transport with periodic flow in homogenous porous media

AbstractThe two-dimensional advection-diffusion equation with variable coefficients is solved by the explicit finitedifference method for the transport of solutes through a homogenous two-dimensional domain that is finite and porous. Retardation by adsorption, periodic seepage velocity, and a dispersion coefficient proportional to this velocity are permitted. The transport is from a pulse-type point source (that ceases after a period of activity). Included are the firstorder decay and zero-order production parameters proportional to the seepage velocity, and periodic boundary conditions at the origin and at the end of the domain. Results agree well with analytical solutions that were reported in the literature for special cases. It is shown that the solute concentration profile is influenced strongly by periodic velocity fluctuations. Solutions for a variety of combinations of unsteadiness of the coefficients in the advection-diffusion equation are obtainable as particular cases of the one demonstrated here. This further attests to the effectiveness of the explicit finite difference method for solving two-dimensional advection-diffusion equation with variable coefficients in finite media, which is especially important when arbitrary initial and boundary conditions are required.

Explicit finite difference solution of the power flow equation in W-type optical fibers

Using the power flow equation, we have calculated spatial transients of power distribution and a steady-state distribution that are due to coupling of guided to leaky modes in W-type optical fibers (doubly clad fibers). A numerical solution has been obtained by the explicit finite difference method. Results show that power distribution in W-type optical fibers depends on both the intermediate layer width and the coupling strength. W-shaped index profile of optical fibers is effective in reducing modal dispersion and therefore in improving the fiber bandwidth. We have also shown that explicit finite difference method is effective and accurate for solving the power flow equation in W-type optical fibers.

Transient Soil Surface Sealing and Infiltration Model for Bare Soil under Droplet Impact

The marked reduction in infiltration rate caused by formation of a soil surface seal due to water droplet impact on bare soil is a well known phenomenon but is rarely considered in infiltration models, especially under center-pivot irrigation. Water application rates under center-pivot irrigation commonly exceed the infiltration rate, especially near the end of the system lateral. This can lead to off-site runoff and erosion, but more importantly results in field-scale non-uniform water infiltration that can substantially reduce water use efficiency of these irrigation systems. The objective of this study was to develop a sealing soil infiltration model that considers transient soil seal formation on a 30 min or less time scale and can potentially be applied to center-pivot sprinkler irrigation systems. A sealing soil infiltration model was developed using an explicit finite difference solution scheme with a transient soil seal formation model, which is unique from other studies in that it explicitly uses droplet specific power as the driving factor for formation of a soil surface seal. The form of the transient seal formation model is also unique in that it is expressed as a rational function of specific power rather than an exponential decay function of droplet kinetic energy. The model was applied to published runoff data from two rainfall simulation studies with varying droplet kinetic energies and application rates on three soils. The sealing soil infiltration model represented the measured infiltration rates very well for all rainfall simulator tests. The transient soil seal formation model uses three parameters, one of which is an empirical parameter representing the susceptibility of the soil to aggregate breakdown that was constant for a given soil. A second model parameter, final saturated hydraulic conductivity of the surface seal, was well correlated to droplet specific power for a given soil. Application of the model to center-pivot irrigation will require the development of a model for estimating droplet specific power and application rate profiles from center-pivot sprinklers for a range of sprinkler designs, flow rates, operating pressures, spacings, and heights.

Numerical Method of Solving Singularity Problems on Potential Computation in Spheroidal Systems

The explicit finite-difference solution method for computing potential distribution within spheroidal systems attracts special difficulties because of the presence of mathematical singularities at the central and pole regions of the spheroids. In this paper, we implement a method for treating the singularities at the pole regions. This entails taking a quarter of the spheroidal cross-section and imposing a Neumann boundary condition along the two lines of symmetry. The results obtained agree perfectly with those obtained from exact solution.

Potential Distribution Computation in Conducting Prolate Spheroidal Shells Using the Exodus Method

This paper presents the use of the Exodus method to compute the potential distribution in a conducting prolate spheroidal shell. The transition probabilities for the Exodus method simulation were obtained from the finite-difference scheme transformation of the prolate spheroid Laplace equation. This method is independent of the random number generation from the computing facility and gives more accurate results than the fixed random walk Monte Carlo method. The results obtained matched perfectly with those obtained by using an exact solution and explicit finite-difference solution method.

Explicit finite difference solution of the diffusion equation describing the flow of radon through soil

Radon diffusion through soil and into air is investigated. The solution of the relevant diffusion equation is given using the explicit finite difference method. Results from a two-medium model (soil-air) are compared to those from a simplified single-medium model (soil alone). The latter are an underestimate in early stages of the diffusion process. Later on, the two models match closely and either one can be used at equilibrium conditions to calculate radon diffusion, estimate indoor radon concentration and assess health hazards.

Solution of the Advection-diffusion Equation using the Differential Quadrature Method

Differential Quadrature Method (DQM) to integrate the one-dimensional Advection-diffusion Equation (ADE) is presented. This method was applied to two examples and the results were compared with the performance of the Explicit Finite Difference Method (EFDM) and Implicit Finite Differences Method (IFDM). Based on the comparison with the exact solution, and both the explicit and implicit finite difference solutions, it was concluded that the DQM provides similar results but less grid points; besides the results are converged quickly. A numerical comparison for a case in which the exact solution is known, DQM gives closer results to the exact values than EFDM and IFDM.

Unconditionally Stable Fully Explicit Finite Difference Solution of Solidification Problems

An unconditionally stable fully explicit finite difference method for solution of conduction dominated phase-change problems is presented. This method is based on an asymmetric stable finite difference scheme for approximation of diffusion terms and application of the temperature recovery method as a phase-change modeling method. The computational cost of the presented method is the same as the explicit method per time-step, while it is free from time-step limitation due to stability criteria. It robustly handles isothermal and nonisothermal phase-change problems and is very efficient when the global temperature field is desirable (not accurate front position). The robustness, stability, accuracy, and efficiency of the presented method are demonstrated with several benchmarks. Comparison with some stable implicit time integration methods is also included. Numerical experiments show that the evaluated temperature field for large values of the Fourier numbers has good/reasonable agreement with the result of small Fourier numbers and the exact/reference solution.

A fast method for numerical simulation of casting solidification

AbstractAn efficient numerical method for simulation of casting solidification is presented. The presented method is based on a stable explicit finite difference solution of the heat equation in conjunction with the temperature recovery method to incorporate latent heat effect. The computational cost of the presented method is approximately the same as an explicit method (per time step), while it is free from the time step limitation due to the stability criterion. A simple domain decomposition method is included to improve the computational performance of the presented method. The efficiency, stability and accuracy of the presented method are supported with illustrative examples. Copyright © 2007 John Wiley & Sons, Ltd.

An enthalpy method for modeling dendritic growth in a binary alloy

An enthalpy fixed grid method is developed for modeling dendritic growth in an under-cooled binary alloy. The proposed numerical method couples explicit finite difference solutions of equations expressing the conservation of enthalpy and solute to an iterative procedure that enforces node by node consistency between enthalpy, solute, liquid fraction, and interface under-cooling. Calculations made with the scheme, are consistent with previously reported work, agree well with limit analytical solutions, approach the correct steady-state tip operating conditions, show grid size independence, are relatively free of grid anisotropy, and can be obtained with a low CPU cost.

A Lagrangian Approach for Oil Recovery in Two-Phase Porous Media Flow

Abstract This paper deals with a fully implicit finite difference scheme for the numerical solution of the Lagrangian form of the porous media fractional flow equation. It covers the theoretical background, description of mathematical formulations, basic assumptions in the models, normalization and transformation between Lagrangian and Eulerian formulations, specification of boundary conditions, discretization and solution method. Some numerical examples are described and the results compared favourably with co-current immiscible displacement data. The solution presented is more efficient than the finite element solution method in terms of requiring less matrix computation effort and it is much more stable than the explicit finite difference solution methods that have been presented in the literature. Due to the ease of formulation and use, the simulation algorithm presented can be used to formulate laboratory numerical simulators that can be used routinely for co-current flow numerical studies. Introduction An equation of the form: Equation (1) (Available In Full Paper) is one of the various Lagrangian forms of the fractional flow equation for two-phase, immiscible displacement in porous media. It is Lagrangian because this configuration allows the equation to be solved as a function of saturation and time. The Eulerian configuration enables the equation to be solved as a function of space and time. Bentsen(1) first derived Equation (1), also called the Bentsen equation(2,3). It was subsequently re-derived by Shen and Ruth(2,3) using a different derivation approach. The equation was derived by combining Darcy's equation for two-phase flow with the continuity equation and by transforming the resulting fractional flow Eulerian form to Lagrangian form. A summary of this derivation is presented later in the paper. This equation, together with normalized forms of the displacement (frontal advance) equation, can be solved to produce a numerical description of a two-phase, incompressible, immiscible displacement process in porous media. The main aim of using the Lagrangian approaches is to enable a more computationally efficient scheme and eliminate the need for space discretization as distance is eliminated from the resulting equation. The form presented in Equation (1) is particularly useful because it is very compact. Theoretical Background Douglas et al.(4) presented an Eulerian technique for solving linear waterflooding problems which involved using a transformed saturation variable and the use of a small mobile water saturation ahead of the flood front to enable the base of the flood front to move forward. Fayers and Sheldon(5) obtained solutions to the one-dimensional displacement equation using both the Lagrangian and the Eulerian approaches. McEwen(6) used the method of characteristics for a numerical solution of the linear displacement equation with capillary pressure. Hovanessian and Fayers(7) also solved numerically the equations describing waterflooding experiments that include both gravity and capillary effects. Their method is similar to that of Douglas et al.(4) because it involves the use of the Eulerian form of the fractional flow equation and the transformation of the resulting second-order non-linear partial differential equations. Hovanessian and Fayers(7) solution differs from those of Douglas et al.(4) because it includes the effect of gravity, allows for the calculation of pressure profiles and makes use of a tabular format for the input of relative permeability and capillary pressure values instead of the u

Unconditionally Stable Fully Explicit Finite Difference Solution of Solidification Problems

An unconditionally stable fully explicit finite difference method for solution of conduction dominated phase-change problems is presented. This method is based on an asymmetric stable finite difference scheme for approximation of diffusion terms and application of the temperature recovery method as a phase-change modeling method. The computational cost of the presented method is the same as the explicit method per time-step, while it is free from time-step limitation due to stability criteria. It robustly handles isothermal and nonisothermal phase-change problems and is very efficient when the global temperature field is desirable (not accurate front position). The robustness, stability, accuracy, and efficiency of the presented method are demonstrated with several benchmarks. Comparison with some stable implicit time integration methods is also included. Numerical experiments show that the evaluated temperature field for large values of the Fourier numbers has good/reasonable agreement with the result of small Fourier numbers and the exact/reference solution.

Numerical errors of explicit finite difference approximation for two-dimensional solute transport equation with linear sorption

The numerical errors associated with explicit upstream finite difference solutions of two-dimensional advection—Dispersion equation with linear sorption are formulated from a Taylor analysis. The error expressions are based on a general form of the corresponding difference equation. The numerical truncation errors are defined using Peclet and Courant numbers in the X and Y direction, a sink/source dimensionless number and new Peclet and Courant numbers in the XY plane. The effects of these truncation errors on the explicit solution of a two-dimensional advection–dispersion equation with a first-order reaction or degradation are demonstrated by comparison with an analytical solution in uniform flow field. The results show that these errors are not negligible and correcting the finite difference scheme for them results in a more accurate solution.

Explicit Finite Difference Solution of Heat Transfer Problems of Fish Packages in Precooling

The present work aims at finding an optimized expli cit finite difference scheme for the solution of problems involving pure heat transfer f rom the surfaces of Pangasius Sutchi fish samples suddenly exposed to a cooling environment. Regular shaped packages in the form of an infinite slab were considered and a generalized mathematical model was written in dimensionless form. An accurate sample of the data set was chosen from the experimental work and was used to seek an optimized scheme of solutions. A fully explicit fin ite difference scheme has been thoroughly studied from the viewpoint of stability, the required time for execution and precision. The characteristic dimension (half thickness) was divided into a numbe r of divisions; n = 5, 10, 20, 50 and 100 respectively. All the possible options of dimension less time (the Fourier number) increments were taken one by one to give the best convergence and t runcation error criteria. The simplest explicit fin ite difference scheme with n = (10) and stability facto r (ΔX) 2 /Δτ = 2) was found to be reliable and accurate for prediction purposes.