Local Grid Refinement Technique Research Articles

Simulation of room flows with small ventilation openings by a local grid-refinement technique

Small ventilation openings in large rooms constitute a numerical resolution difficulty in indoor air flow simulation. A local grid-refinement technique is introduced to add local grid points in the small-opening regions. The method requires a relatively small increase in computer storage. A multi-grid algorithm, which turns out to be an efficient and accurate solves is used to solve the discrete system of equations on both global and local grids. A method for implementing the wall function in the solution procedure is described. Solutions of a three-dimensional isothermal mixing ventilation problem at Reynolds number of 5000 are compared with the experimental results in the literature. The method is then applied in a three-dimensional scaled model room ventilated by a plaque diffuser. The calculations show good agreement with the smoke visualisations of the flow pattern close to the diffuser. The method can be used to predict air flows with more realistic ventilation openings and more applications in large enclosures are expected in the future.

Calculation of turbulent combustion of propane in furnaces

AbstractAn evaluation of some numerical methods for turbulent reacting flows in furnace‐like geometries is carried out. The Reynolds averaged Navier–Stokes equations and the two‐equation k–ϵ model together with either finite‐rate or infinite‐rate reaction models are solved numerically. Either single‐ or multiple‐step reactions together with the ‘eddy dissipation concept’ (EDC) are used to model reacting flows with finite reaction rates. The numerical scheme is finite difference based, together with a multi‐grid method and a local grid refinement technique. These methods have been used to calculate the combustion of propane in a single‐ and multiple‐burner configurations. In the former case, the sensitivity of the solution to variations in some model parameters (determining the reaction rate) and numerical parameters (mesh spacing) has been studied. It is noted that different dependent variables exhibit different levels of sensitivity to the variation in model parameters. Thus, calibration and validation of models for reacting flows require that one compares the most sensitive variables. For engineering purposes, on the other hand, one may calibrate and validate models with respect to the most relevant variables. Our conclusion is that since sensitivity of the temperature distribution is relatively mild, one can still use EDC‐like methods in engineering applications where details of the temperature field are of minor importance.

Calculation of a Confined Axisymmetric Laminar Diffusion Flame Using a Local Grid Refinement Technique

Abstract In this paper a laminar diffusion flame is predicted using a local grid refinement technique and non-staggered meshes. A single coarse grid covers the whole domain and local grid refinement is carried out in the regions of high gradients without changing the basic grid structure. The Navier-Stokes, the energy and the chemical species transport equations are solved and a finite rate chemical kinetics combustion model is employed. Transport equations for the species concentration are solved using a pseudo time splitting technique. The predictions are in good agreement with the experimental measurements for a confined laminar methane flame. The numerical method proved to be robust and accurate.

Characteristic, local grid refinement techniques for reservoir flow problems

AbstractUsing a sequential time‐marching approach, we describe and implement effective solution strategies for the equations governing immiscible displacement in a porous medium. By combining adaptive grid refinement with an operator‐splitting technique based on the modified method of characteristics, the saturation equation is treated in a consistent and accurate way. For the pressure equation we construct accurate piecewise linear velocity components from a piecewise linear pressure approximation.

Development and Testing of a Static/Dynamic Local Grid-Refinement Technique

Summary Simple, efficient, and stable static and dynamic local grid-refinementprocedures for multi-dimensional, multiphase reservoir problems are developedand tested problems are developed and tested by isothermal reservoirsimulation. Introduction There has been great interest in reducing excessive computer usage innumerical modeling without sacrificing the degree of accuracy. To achieve this, various mesh-refinement methods have been developed that use denser gridpointsonly in localized regions where truncation errors may be fatal. The physical domain of interest is subdivided by equally or unequally spacedgrid-points to form a computational domain. A grid model is the geometricrepresentation of the resulting computational domain. A coarse- (or base) gridmodel represents the preliminary subdivision of the physical preliminarysubdivision of the physical domain with a grid spacing of (Fig. 1a). The basegrid model can be refined by reducing A. A fine-grid model is formed byreducing A in the entire domain, as shown in Fig. 1b. For a conventionallyrefined grid model, the grid spacing is reduced only within a sub-region of thecoarse model. However, the fine mesh lines are extended to the boundaries ofthe physical domain (Fig. 1c). A locally refined grid model is a special caseof a conventional] refined model. In this case, the fine mesh lines are notextended to the exterior boundary of the reservoir (fig. 1d). For dynamically refined grid models, the number of gridpoints istime-dependent, and the locally refined region varies spatially. Composite grid(also known as hybrid grid) models combine cylindrical (or elliptical) andrectangular grid systems. They have been used extensively in studies ofnumerical heat transfer and in fluid-mechanics problems. Multigrid models usecoarse- and problems. Multigrid models use coarse- and fine-grid networkssequentially in the entire computational domain. Static grid-refinement studiesfocused on developing techniques for grid interactions at the periphery of thecoarse and fine grids. Lam and Simpson developed a local mesh-refinement technique for numericalsolution of advection-diffusion transport equations. Graham and Smart describeda reservoir simulator that uses a fine-grid model nested in a coarse-grid modelfor areal simulation representing a reservoir in communication with a largepressure-supporting aquifer. In 1982, Rosenberg developed a mesh-refinementstrategy similar to Lam and Simpson's technique. Quandalle and Besset studiedthe efficiency of the grid-refinement technique described by Rosenberg bysimulating displacement of oil by water in a confined five-spot pattern. In1983, Heinemann et al. developed a procedure for dynamic and local gridprocedure for dynamic and local grid refinement in conjunction with amultiple-application reservoir simulator. Forsyth and Sammon generalized Heinemann el al.'s grid-refinement technique for modeling faults andpinchouts. Heinemann et al. described a dynamic grid-refinement method as an addendumto their static grid-refinement scheme. Coarse- and refined-grid interactionsat the periphery of the two grid systems are similar to those in Heinemann etal.'s static grid-refinement technique. Additionally, specially developed datamanagement for dynamic subdivision is needed. They used preset limiting valuesfor saturatoin, composition, or temperature changes when subdividing the -basegrid. Han et al. modified and improved Heinemann et al.'s dynamicgrid-refinement technique. Pedrosa and Aziz developed a procedure to improve wellblock calculations byprocedure to improve wellblock calculations by combining an orthogonalcurvilinear grid network with a rectangular grid network in the vicinity of awell point. They compared the performance of their hybrid grid against acoarse-grid model using waterflooding and water-coning problems. Brand's multigrid technique has been used extensively for a wide range ofproblems. The multigrid method uses two problems. The multigrid method uses twogrid patterns, one fine and one coarse, that sequentially envelope the entireregion. Basically, in Brandt's technique, a preliminary solution is obtained inthe coarse-grid preliminary solution is obtained in the coarse-grid domain withthe aid of a predetermined iteration criterion. Then, information istransferred from the coarse grid to the fine grid by interpolation of thissolution. The iteration is continued with the fine-grid model. After theiteration is satisfied, the improved solution and residual of the fine grid aretransferred back into the base grid. JPT P. 487

Fluid dynamics of a stationary weld pool

Flow driven by a combination of thermocapillary, Lorentz, and buoyant forces has been investigated in an axisymmetric and stationary weld pool numerically. By assuming a small value for the capillary number, the top and bottom boundaries can be taken to be flat, and the surface deflections can be calculateda posteriori as a domain perturbation. Owing to thin boundary layers that exist at the top free surface and next to the vertical wall, very fine grids are required in these regions in order to obtain an accurate solution to the Boussinesq form of the Navier-Stokes equations. This was done by solving the governing equations by multigrid methods to which a local grid refinement technique was added. Welding of both aluminum and steel were considered. The essential difference between these two materials for this analysis is that the Prandtl number of aluminum is an order of magnitude smaller than that of steel. Through a parametric study, the thermocapillary forces and Lorentz forces were found to dominate buoyancy forces in a typical welding situation. Although the flows in weld pools include a pronounced recirculating region near the top surface, isotherms could be determined in the case of aluminum to a good approximation by a conduction analysis, owing to the smallness of its Prandtl number and the relative thinness of the welded plate considered. For steel, the isotherms deflect considerably for high current inputs.

Preconditioned conjugate gradient methods for large-scale fluid flow applications

The simulation of large-scale fluid flow applications often requires the efficient solution of extremely large nonsymmetric linear and nonlinear sparse systems of equations arising from the discretization of systems of partial differential equations. While preconditioned conjugate gradient methods work well for symmetric, positive-definite matrices, other methods are necessary to treat large, nonsymmetric matrices. The applications may also involve highly localized phenomena which can be addressed via local and adaptive grid refinement techniques. These local refinement methods usually cause non-standard grid connections which destroy the bandedness of the matrices and the associated ease of solution and vectorization of the algorithms. The use of preconditioned conjugate gradient or conjugate-gradient-like iterative methods in large-scale reservoir simulation applications is briefly surveyed. Then, some block preconditioning methods for adaptive grid refinement via domain decomposition techniques are presented and compared. These techniques are being used efficiently in existing large-scale simulation codes.

Finite element techniques for convective-diffusive transport in porous media

Operator-splitting techniques are applied to convective-diffusive transport problems in porous media. The convection is treated by applying a modified method of characteristics to time-step along the characteristics of the convective part of the flow. The nonsymmetry in the spatial operator is addressed via a Petrov-Galerkin method which uses a test function to achieve stability through a balancing of the remaining convection, the diffusion, and any possible reaction terms. The use of time-stepping along characteristics allows the use of large time-steps in a stable but accurate fashion. If local phenomena are important, self-adaptive local grid refinement techniques can be coupled with the operator splitting.

Adaptive grid refinement methods for time‐dependent flow problems

AbstractIn the multi‐dimensional numerical simulation of certain multi‐phase fluid flow processes, many phenomena are sufficiently localized and transient that self‐adaptive local grid refinement techniques are necessary to resolve the local physical behaviour. For large‐scale simulation problems, efficiency is the key to the choice of specific adaptive strategies. Several different types of self‐adaptive methods will be discussed and briefly compared. The differences in complexity of the data structure and efficiency of the solution techniques will be explained. Concepts for incorporating these methods in existing large‐scale simulators will also be presented. Man man‐years have been invested in developing large‐scale numerical simulators which do an excellent job of resolving most of the larger length‐scale phenomena in our production processes. Although adaptive local refinement methods are critical for many applications, most techniques which have been developed to date either cannot be incorporated in our codes or are so computationally complex that they destroy the efficiency of existing simulators. Recently, new preconditioning techniques have been developed which allow grid refinement methods to be used efficiently in exisiting codes without disrupting the basic solution process. These techniques are highly amenable to vectorization and parallelization of the algorithms. They are also a first step in allowing complex well models to be incorporated into large field‐scale codes and applications.

Efficient adaptive procedures for fluid-flow applications

In the multi-dimensional numerical simulation of certain multi-phase fluid-flow processes, many phenomena are sufficiently localized and transient that self-adaptive local grid-refinement techniques are necessary to resolve the local physical behavior. For large-scale simulation problems, efficiency is the key to the choice of specific adaptive strategies. For example, many techniques which are excellent for smaller, time-independent processes are very inefficient for large-scale transient modeling. Several different types of self-adaptive methods are discussed and briefly compared. The differences in complexity of the data structure and efficiency of solution are explained. Certain preliminary numerical results are discussed.

Finite element methods for nonlinear flows in porous media

A model problem is presented for fluid flow in porous media which has numerical difficulties common with many other areas of nonlinear mechanics. Special finite element techniques are described which have proved useful in alleviating many of these difficulties. A modified method of characteristics, mixed finite element methods for fluid velocities, singularity removal, and self-adaptive local grid refinement techniques are presented. Both theoretical and computational results are described, together with an indication of the present and future uses of finite element methods in large-scale reservoir simulation.