Fixed-grid Method Research Articles

PREDICTING NATURAL-CONVECTION-DOMINATED PHASE CHANGE PROBLEMS BY CONTROL VOLUME UNSTRUCTURED TRIANGULAR GRID: APPLICATIONS TO THE MELTING OF PURE METAL

Control volume methods have recertify been developed for fluid flow and heat transfer on unstructured meshes. In this study, we extend these methods to implement the solution of natural-convection-dominated milting of gallium by a fixed-grid method. A simple, robust, and reliable explicit numerical method ( MAC method) is applied for an unstructured triangular grid. This investigation also applies the implicit SIMPLER method for an unstructured triangular grid. Results obtained from the unstructured triangular grid correlate well with the structured mesh computations and experimental data. Also, the feasibility of applying the triangular grid to complex geometric problems is demonstrated by calculating two different triangular domains.

CYCLIC MELTING AND SOLIDIFICATION OF STEEL

A numerical investigation is conducted to study the effect of cyclic wall temperature on the melting and solidification of steel in a rectangular enclosure. The formulation is based on the fixed-grid enthalpy method and the SIMPLE algorithm. It was found that the low-carbon steel shows a superior thermal behavior. In the steady-periodic situation the period of oscillations of the liquid curves and that of the wall temperature are equal, but the liquid curves are lagging in phase. This study also revealed the main flow features of the cyclic phase-change processes with and without the mushy region.

Flexible Discretization Algorithm for Fixed-Grid MOC in Pipelines

The problem of selecting a time step that exactly satisfies the Courant condition arises most commonly when the fixed-grid method of characteristics (MOC) is applied to multipipe systems. Traditionally, this problem has been solved by employing one of a variety of interpolation techniques or by allowing small adjustments in the value of wave speed. In this paper, the number of available strategies is increased substantially by allowing several new kinds of interpolation as well as blended combinations of these interpolation approaches with the wave-speed–adjustment technique. The new hybrid approaches include interpolation along a secondary characteristic line and minimum-point interpolation, which minimizes the distance from the interpolated point to the primary characteristic. An intuitively appealing technique involving direct adjustment of the wave path is rejected because it is potentially unstable. The entire flexible algorithm is implemented as a preprocessor step, uses memory efficiently, executes quickly, and provides a flexible tool for investigating the importance of discretization errors in pipeline systems. The properties of the new algorithm are analyzed theoretically and illustrated by example.

Variable grid decomposition for ill-posed interferometric tomography: Comparison of algorithms

An algorithm, termed the variable grid method (VGM), has been developed to provide enhanced reconstruction by interferometric tomography under the ill-posed problems of restricted angular scanning and incomplete projection that frequently arise in flow measurements. The algorithm is based on variable grid decomposition of a field. The performances of the two versions of the VGM are compared to investigate the effects of the redundancy criteria in generating variable grid elements. Performance comparison is also made with a conventional fixed grid method. The testing is conducted through computer simulation of experiments and reconstruction with a known three-dimensional field that has been produced by computational fluid dynamics. The reconstruction through computer simulation demonstrates the superiority of the two VGMs to the conventional method. Both the maximum and average reconstruction errors are reduced appreciably. The VGM version with stricter parameter contro performs best. The technique can reflect spatial information contained in measured data. Thus, improvement is noticeable in the regions with dense scanning of probing rays. These regions are usually of interest in engineering applications.

Efficient Valve Representation in Fixed-Grid Characteristics Method

This work formulates a unified set of boundary conditions to efficiently represent the majority of valve and orifice devices found in water supply, transmission, and distribution systems. A particularly useful combination of mathematical components results when a lumped inertia model is linked with a throttling device. This combination of elements, termed a pipe replacement element/valve-in-line (PREVIL), has been constructed to permit a wide range of control-valve/short-pipe combinations to be conveniently modeled with the method of characteristics. The solution is quadratic in form and explicit, regardless of the number of pipes that are connected to the boundary condition. A variety of on-off and modulating valves can be accurately handled within this framework. An additional feature developed in this treatment of pressure-reducing an pressure-sustaining values, flow controllers, and other similar devices permits a more realistic representation of these important control devices. The response time of the regulating valve on opening or closing can be readily approximated so that a greater range of control behavior can be simulated. Application of the combined boundary condition is illustrated in a pipe network.

Analytical and numerical solutions of fixed and moving boundary problems for liquid surfactant membrane processes

A mathematical model describing a batch liquid-surfactant-membrane (LSM) process is developed. In addition to unsteady-state diffusion, reaction, and external mass transfer, consumption of the reagent by the reaction is also considered. Thus, depending on the Thiele modulus and the initial amount of the reagent, the model either remains to be a fixed boundary problem throughout the whole process or changes to a moving boundary problem after some initial period of fixed boundary. For the fixed boundary problem a general analytic solution is obtained in an eigenfunction expansion by a self-adjoint formalism in linear operator theory. In contrast tot he known perturbation solution, the solution is exact and straightforward to use. For typical operations the series solution converges rapidly within a few terms, providing a useful tool in design and analysis of the LSM process. Also the solution readily enables determination of the time of transition from fixed to moving boundary. For the moving boundary model, a simple numerical method based on a fixed-grid finite difference method is constructed for solution. In this method the position of the moving boundary is approximated to a grid point at which depletion of the reagent occurred most recently. The calculated position of the moving boundary has been found to be accurate to the size of the grid. The method has also been found to be stable and reliable for all practical ranges of its application.

NATURAL CONVECTION DURING MELTING AND SOLIDIFICATION OF PURE METALS IN A CAVITY

An enthalpy-porosity fixed-grid method has been applied to the melting and solidification of pure metals in a rectangular cavity. Flow structures and isotherms are compared with literature information based on multidomain analysis. During solidification, recirculation cells are observed to vanish with time, whereas the multidomain method predicted their enlargement. The cell near the interface significantly affects the local solidification rate at the cavity top. Results of the present study agree well with the experimental data in the literature. The effect of neglecting the viscous diffusion term for a phase change problem involving low Prandtl number liquid metals is also studied. The maximum value of the stream function is higher by 10% and the melt volume by 5%, in the absence of viscous effects.

TRANSIENT ANALYSIS OF THE START-UP OF A WATER HEAT PIPE FROM A FROZEN STATE

A two-dimensional transient model is developed to simulate the start-up of a radiatively cooled water heat pipe from a frozen state. The model incorporates sublimation and resolidification of working fluid; melting of working fluid in the porous wick structure; condensation of working fluid as a thin film on a frozen substrate, and drainage of the film; liquid flow and heat transfer in the porous wick; and thermal and hydrodynamic couplings of phases at their respective interfaces. The model handles the change of phase in the wick using a modified fixed-grid homogeneous enthalpy method, employing a mushy-cell temperature range of 2 × 10−8 K (limited by machine accuracy), without numerical instabilities. The simple method, based on the frozen fraction developed to calculate the heat fluxes at the boundaries of the mushy cell, improved accuracy and reduced oscillations in temperature time histories by 1–2 orders of magnitude.

NUMERICAL SOLUTION OF PLANE-FRONT SOLIDIFICATION WITH KINETIC UNDERCOOLING

The numeric solution of a modified Stefan problem in the presence of a generalized kinetic undercooling term is investigated. Two fixed-grid methods and a deforming-grid scheme are described. The fixed-grid phase-field method is extended to encompass a generalized dependence of the melting temperature on the front velocity. Two liquid fraction-based methods are also provided, which are discretized over a deforming grid and a fixed grid, respectively. Results show that, whereas the deforming-grid scheme is the most efficient for one-dimensional problems, the phase-field method is a competitive choice among fixed-grid schemes.

Experimental and numerical comparison of extractive and in situ laser measurements of non-equillibrium carbon monoxide in lean-premixed natural gas combustion

A two-dimensional transient fixed-grid enthalpy-based numerical method is developed to analyze the effects of turbulent transport during a binary alloy solidification process. Turbulence effects are introduced through standard k–ε equations, where coefficients are appropriately modified to account for phase-change. Microscopically-consistent estimates are made regarding temperature-solute coupling in a non-equilibrium solidification situation. The model is tested against laboratory experiments performed using an NH4Cl–H2O system in a rectangular cavity cooled and solidified from the top. Particular emphasis is laid on studying the interaction between Rayleigh–Benard type convection and directional solidification in the presence of turbulent transport. Numerical predictions are subsequently compared with experimental results regarding flow patterns, interface growth and evolution of the temperature field, and the agreement is found to be good.

A fixed grid method for the numerical solution of phase change problems

AbstractA fixed grid method using an updated iterative implicit scheme is developed to solve one‐dimensional phase change problems. The temperature field is deduced from the resolution of the governing equations whose discretization takes into account the discontinuous variation of the temperature derivative at the phase change front. At each iteration an updated position of the moving front is found from the resolution of the energy conservation at the solid‐liquid interface. The accuracy of the proposed numerical method has been checked on three test problems.

Stability analysis of Eulerian-Lagrangian methods for the one-dimensional shallow-water equations

In this paper stability and error analyses are discussed for some finite difference methods when applied to the one-dimensional shallow-water equations. Two finite difference formulations, which are based on a combined Eulerian-Lagrangian approach, are discussed. In the first part of this paper the results of numerical analyses for an explicit Eulerian-Lagrangian method (ELM) have shown that the method is unconditionally stable. This method, which is a generalized fixed grid method of characteristics, covers the Courant-Isaacson-Rees method as a special case. Some artificial viscosity is introduced by this scheme. However, because the method is unconditionally stable, the artificial viscosity can be brought under control either by reducing the spatial increment or by increasing the size of time step. The second part of the paper discusses a class of semi-implicit finite difference methods for the one-dimensional shallow-water equations. This method, when the Eulerian-Lagrangian approach is used for the convective terms, is also unconditionally stable and highly accurate for small space increments or large time steps. The semi-implicit methods seem to be more computationally efficient than the explicit ELM; at each time step a single tridiagonal system of linear equations is solved. The combined explicit and implicit ELM is best used in formulating a solution strategy for solving a network of interconnected channels. The explicit ELM is used at channel junctions for each time step. The semi-implicit method is then applied to the interior points in each channel segment. Following this solution strategy, the channel network problem can be reduced to a set of independent one-dimensional open-channel flow problems. Numerical results support properties given by the stability and error analyses.

Efficient numerical methods for infiltration using Richards' equation

Two efficient finite difference methods for solving Richards' equation in one dimension are presented, and their use in a range of soils and conditions is investigated. Large time steps are possible when the mass‐conserving mixed form of Richards' equation is combined with an implicit iterative scheme, while a hyperbolic sine transform for the matric potential allows large spatial increments even in dry, inhomogeneous soil. Infiltration in a range of soils can be simulated in a few seconds on a personal computer with errors of only a few percent in the amount and distribution of soil water. One of the methods adds points to the space grid as an infiltration or redistribution front advances, thus gaining considerably in efficiency over the other fixed grid method for infiltration problems. In 17‐s computing, this advancing front method simulated infiltration, redistribution, and drainage for 50 days in an inhomogeneous soil with nonuniform initial conditions. Only 16 space and 21 time steps were needed for the simulation, which included early ponding with the development and dissipation of a perched water table.

On time dependent multi-grid numerical technique

The multi-grid method which employs a sequence of nested grids in the solution process is a general numerical technique for solving continuous problems. Implementation of multi-grid algorithm can result in higher accuracy and efficiency than methods with the fixed-grid size and fixed-order of approximation, and specific tests of this are considered in the present study. An introduction to the implementation of the multi-grid method in the solution of time dependent partial differential equation is presented. Multi-grid solutions for the advection equation are compared with published results using the usual fixed-grid method.