Non-orthogonal Grids Research Articles

Efficient and accurate FDTD algorithm for the treatment of curved material boundaries

The application of finite-difference time-domain methods to the analysis of structures having curved boundaries is a long-standing problem. Traditionally, either the staircasing approximation which is simple but inaccurate, or the generalised nonorthogonal grids which are accurate but complex have been used. In this contribution a novel approach is presented which combines the efficiency of the Cartesian mesh with the accuracy of the conformal grid. Results are presented for some example structures which are in good agreement with other methods.

Thermal effects of acrylic cementation at bone tumour sites

The use of acrylic bone cement as an adjunct to surgical excision of giant cell tumour of bone appears to reduce the incidence of tumour recurrence. Possible mechanisms for this apparent tumour inhibition include cytotoxic effects from the methylmethacrylate monomer and tissue hyperthermia from the heat of polymerization of the cement. This work presents a method for the prediction of temperature fields and resulting tissue necrosis arising from the implantation of polymethylmethacrylate (PMMA) at the site of a curretted giant cell tumour of bone. This is accomplished using a two-dimensional model based on geometry obtained from digitized MRI images of the distal femur. A general-coordinate, non-orthogonal grid generation technique is used and solutions are obtained with an alternating-direction implicit (ADI) finite-difference scheme. The nodal temperature histories are then used to evaluate the effect of variable defect size on the zone of thermally induced cell necrosis. The results suggest the depth of the necrotic region is quite sensitive to the size of the implant. In at least some cases, the heating effect is sufficient to cause significant necrosis of tumorigenic cells. Implanting a large mass of acrylic may risk overkill, damaging substantial amounts of healthy tissue.

The effect of non-orthogonal grids on spray and air flow predictions

This paper describes the effect of non-orthogonal grids on both spray and air flow predictions, and the effect of higher order convection scheme on inclined grids. In order to assess the effect of non-orthogonality of grids a number of different grid cases are generated, and applied to the same conditions of injected sprays. The effects on the free/wall spray penetration and Sauter mean diameter (SMD) are discussed. The results show that the numerical error is increased with increasing non-orthogonality, and that Hybrid is better than QUICK for the calculation of sprays on inclined grids. For the calculation of the spray injection, which has always large gradient regions at the front end, adaptive grids following the direction of spray/ gas flow might be required.

OVERLAPPING CONTROL VOLUME APPROACH FOR CONVECTION-DIFFUSION PROBLEMS

This paper introduces a finite volume method to solve 2D steady state convection–diffusion problems on structured non-orthogonal grids. Overlapping control volumes (OCV) are used to discretize the physical domain and the governing equations are solved without transformation. An isoparametric formulation is used to compute diffusion and for upwinding. Four test problems are solved using this and other schemes. The modelling of diffusion in OCV seems very effective even on distorted meshes. The convection modelling in OCV is found to be second-order-accurate, like QUICK, on regular meshes. Although its accuracy is slightly inferior to the latter on rectangular grids, its faster convergence gives it a better overall performance. On non-orthogonal grids, OCV gives better accuracy for a large and practical range of Peclet numbers than does QUICK applied to the transformed equations using the conventional five-point diffusion modelling. The results obtained also demonstrate that the scheme reduces false diffusion to a considerable extent in comparison with the power-law scheme.

Assessment of grid interface treatments for multi-block incompressible viscous flow computation

In the multi-block computation of the Navier-Stokes equations, the interface treatment is a key issue. In the present work, we investigate this issue in the context of a pressure-based method using a non-orthogonal grid. For the momentum equations, a straightforward bilinear interpolation seems satisfactory as the interface treatment; on the other hand, because the pressure field depends on the satisfaction of the mass continuity equation, a conservative interface treatment has been found necessary for the pressure-correction equation. Two alternative interface treatments for the pressure-correction equation, one employing the Neumann boundary condition in both grid blocks, based on explicit local, cell-by-cell mass flux conservation, and the other utilizing Neumann-Dirichlet boundary conditions, allowing the interface condition in one block to be derived by interpolating the pressure field from the adjacent block, are assessed in the present work. To evaluate these interface schemes, the laminar flow inside a lid-driven cavity flow, and the turbulent flow around cascades of multiple airfoils have been investigated. For the case tested, both interface treatments give comparable accuracy. The finding that more than one type of interface treatment can work well allows one to devise a flexible multi-block strategy for complex flow computations.

A MULTIGRID ALGORITHM FOR A MULTIBLOCK PRESSURE-BASED FLOW AND HEAT TRANSFER SOLVER

A multigrid algorithm is developed along with an implicit multiblock pressure-based solver for calculating flow and heat transfer problems on nonorthogonal grids. The implicit treatment of the block interface has proven to be important for the efficiency of a multiblock method. In this study, the implicitness is adopted for all grid levels during a multigrid solution process. As a result, the block interface becomes invisible in the final solutions and the convergence is not adversely affected. The proposed multiblock multigrid algorithm is demonstrated on several representative two-dimensional flow and heat transfer problems. The results show that order-of-magnitude saving in computational time can be achieved with the multigrid algorithm.

Multidimensional Water Flow in a Low‐Level Waste Isolation Barrier

AbstractAn analysis was conducted of multidimensional subsurface moisture flow in a hypothetical low‐level radioactive waste disposal facility. The key design feature examined in the analysis was a sloping sand/gravel capillary barrier designed to route natural infiltration around a concrete vault. Three barrier slopes (1:25, 1:10, and 1:5) and three sand/gravel property combinations were considered. The slopes and material property combinations were selected to represent a range in barrier effectiveness. The porous media flow code used for this analysis was VAM3DCG, a three‐dimensional, finite‐element code which was able to use nonorthogonal grid discretizations and employed robust, efficient numerical techniques. Three‐dimensional modeling demonstrated that flow in the hypothetical design exhibited cross‐slope flow because of the pressure gradient produced in the third, cross‐slope dimension. Barrier effectiveness was shown to be highly sensitive to the sand/ gravel material properties. Barrier slope was less important, especially for the effective material combinations. The presence of three‐dimensional flow could be important in a performance assessment if the quantity of water predicted to breach the capillary barrier by a two‐dimensional model were different from that predicted by a three‐dimensional model. Comparative modeling demonstrated that a two‐dimensional analysis resulted in underestimation of barrier effectiveness. For the considered design, two‐dimensional modeling is a conservative, yet reasonable, approach in a performance assessment application.

An Investigation of Combustion Chamber Shapes for Small Automotive Direct Injection Diesel Engines Employing Spray Impaction

In modern compact combustion chambers of direct injection diesel engines with high-speed injection spray, the impaction of the spray on the wall is almost inevitable. This may result in high levels of unburnt hydrocarbons. However, a number of combustion systems have been developed which deliberately employ wall impaction as a means of breaking up the fuel spray and/or directing it in a desired direction. These include chambers which employ impaction sites attached to the cylinder head or a cut-off pip in the piston bowl centre. A further type of geometry, which has been the subject of an earlier investigation by the authors, is presented here, in which cut-off pips are provided for a number of sprays directed not only straight down into the combustion chamber centre but also across and down into the bowl in the normal radial direction. For the current analysis, the sizes of the pips and their positions are based on the previous work. All these cases are investigated using a computation fluid dynamics code employing non-orthogonal grid systems, the k–epsis; turbulence model, the implicit PISO {pressure implicit by splitting of operators) solution scheme, and the discrete droplet model for the sprays. Comparisons of simulations with relevant experimental data show that the code returns good qualitative and quantitative results. Apparently good qualitative results are also returned for a number of the combustion chambers employing impaction sites attached to the combustion head and in the centre of the piston bowl. Simulations of combustion chambers employing a number of impaction sites around the piston bowl indicate the likely advantages of such a system over a conventional highly swirled one.

A staggered-grid finite volume method for the vorticity-velocity equations

This paper provides a novel finite volume technique for solving two-dimensional viscous steady flows using the vorticity-velocity equations written as a second-order system. The parabolic vorticity transport equation and the elliptic equations for the velocity components are discretized in time using a two level implicit Euler scheme and in space using a staggered-grid finite volume discretization. A line-Gauss-Seidel iteration combined with a deferred-correction procedure is used to drive to zero the steady-state residuals, which are second-order accurate and satisfy exactly the finite volume discrete forms of the continuity equation and of the vorticity definition. The accuracy and general applicability of the method are demonstrated vs a few model problems using orthogonal as well as nonorthogonal grids.

Acceleration of iterative algorithms on highly clustered grids

A family of new methods has been developed to accelerate the convergence rate of iterative algorithms for obtaining a steady-state solution as an asymptotic limit of an unsteady second-order partial differential equation or a system of such equations. It was assumed that a central differencing has been used for spatial discretization. The new acceleration methods are based on the sensitivity of the future residual at every grid point to the change in the solution vector components at the neighboring grid points used in the local discretization approximation. The acceleration parameters introduced in the methods have been optimized with the objective to minimize the future global residual. The new sensitivity-based methods have been applied to finite difference codes for twoand three-dimensional, laminar, incompressible flow Navier-Stokes equations; two-dimensional, turbulent, incompressible flow Navier-Stokes equations; and two-dimensional, compressible flow Euler equations. The new sensitivity-based acceleration methods demonstrated superior performance in all test cases that involved severe grid clustering and grid nonorthogonality and included laminar and turbulent flows with closed and open flow separation.

The finite volume flic method and its stability analysis

In this study, a general finite volume Fluid-in-cell method (FVFLIC) for solving the Navier—Stokes equations is introduced. The stability of the numerical method is then analysed by directly using two-dimensional Euler equations instead of a linear model equation. This direct approach to the analysis of non-linear stability is based on the Taylor expansion of the discretized Euler equations and some basic principals that have been used for analysing linear model equations. The exact forms of numerical viscosity or truncation errors are derived and discussed. The influences of the numerical viscosity as well as the artificial viscosity on numerical solutions are investigated. Results from this analysis can be used to construct appropriate artificial viscosity terms. Based on the above methodology, a stability criterion is proposed for the calculation of time steps for general three-dimensional computation using non-orthogonal grids.

An Analysis of the Robustness of Some Incomplete Factorizations

The performances of various incomplete-factorization preconditioners for conjugate-gradient-like methods have been investigated by several authors, using numerical experiments. Recently it has been suggested that the number of iterations is almost directly related to the norm of the error matrix. In this paper, a simple method to obtain analytical expressions approximately proportional to the norm of the error matrix is suggested. These expressions can be used to analyse the relative performances of different incomplete factorizations. Here, they are employed to analyse the performances of the ILU and INV(1) algorithms (including a slight generalization of the latter algorithm, to cover the case of nonorthogonal grids as well) when applied to two-dimensional problems with different characteristic properties, including numerical anisotropy, large convection terms, and nonorthogonal grids. That is, the robustness of these factorizations is analysed. The results from the analyses are compared to results obtained from numerical experiments on problems with the same characteristic properties. Both analyses and numerical experiments show that INV(1) is robust with respect to varying problem characteristics, while ILU is not. ILU is very dependent upon a good grid-node ordering for each problem. The results from the analyses give some guidance in this respect.

Assessment of the SSG pressure-strain model in two-dimensional turbulent separated flows

Data from a number of benchmark two-dimensional turbulent separated flows are used to assess the performance of the Speziale, Sarkar and Gatski model [28] for the fluctuating pressure-strain correlations and, in particular, to determine whether the model can reproduce the effects of rigid boundaries without recourse to ad-hoc corrections. Comparative predictions are also obtained with a standard Reynolds-stress closure in which such correlations are necessary and these serve to distinguish the models' true performance from spurious computational artifacts. The models were implemented in a finite-volume method which solves the Navier-Stokes equations on non-orthogonal grids with colocated storage arrangement. Details of the numerical treatments adopted to obtain stable and fast-converging solutions are described and the outcome of rigorous grid-independence checks are reported for each test case. It is found that both models yield essentially similar results for the mean flow and turbulence fields. The implications of this result to the choice of Reynolds-stress closure for complex engineering simulations are discussed.

Three-dimensional numerical modelling of water flow in a river with large bed roughness

A numerical model for three-dimensional simulation of water flow in rivers with large roughness elements is developed. The bed roughness elements are large stones and rocks in the rivers. The model uses a finite volume method to solve the Navier-Stokes equations for three dimensions on a general non-orthogonal grid. The k-e turbulence model is used to solve the Reynolds-stress term. A porosity model is developed to model the bed roughness elements. The porosity model is used in combination with the solution of the Navier-Stokes equations to calculate interactions between porous and non-porous areas. To test the model, a reach of the Norwegian river Sokna is modelled. Velocity measurements from the river are taken at a number of locations and at several discharges. The measured velocities compare well with the results from the numerical model.

A high resolution and bounded convection scheme

A high resolution and bounded convection scheme is proposed for the simulation of steady incompressible flows with finite volume method. The scheme is formulated on a nonuniform, nonorthogonal grid so as to be applicable to the simulation of practical engineering problems. The relative performance of the scheme is evaluated through applications to the test problems. The results of numerical experiments show that the proposed scheme yields similiar solutions which are as good as those obtained with the QUICK scheme, but without exhibiting the physically unrealistic overshoots and undershoots.

Techniques and results of tokamak-edge simulation

This paper describes recent development of the UEDGE code in three important areas. (1) Non-orthogonal grids allow accurate treatment of experimental geometries in which divertor plates intersect flux surfaces at oblique angles. (2) Radiating impurities are included by means of one or more continuity equations that describe transport and sources and sinks due to ionization and recombination processes. (3) Advanced iterative methods that reduce storage and execution time allow us to find fully converged solutions of larger problems (i.e., finer grids). Sample calculations are presented to illustrate these developments.

A comparison of higher-order bounded convection schemes

This paper presents a comparison of recently developed four higher-order bounded convection schemes SOUCUP, HLPA, SMARTER and LPPA which employ different characteristics in a normalized variable diagram. All the schemes are formulated on a non-uniform, non-orthogonal grid and their relative performances are compared through applications to the several test problems. The results of numerical experiments show that all the tested bounded schemes resolve the boundedness problem retaining the accuracy of the higher-order scheme. The HLPA, SMARTER and LPPA schemes result in nearly the same solution behaviours and the SOUCUP scheme is shown to be relatively more diffusive than the other three bounded schemes.

Application of wall functions to generalized nonorthogonal curvilinear coordinate systems

A method has been developed for the application of wall functions to generalized curvilinear coordinate systems with nonorthogonal grids. Two test cases have been computed using this method with the k-e turbulence model: flow over a flat plate at 0-deg angle of attack using a nonorthogonal grid at the wall and flow over a prolate hemispheroid with a hemispherical nose cap at 0-deg angle of attack. All results are compared with experimental data. In addition, the hemispheroid results are compared with computations using the Baldwin-Lomax algebraic turbulence model and the Chien low-Reynolds-number k-e turbulence model.

A multigrid solver for fluid flow and mass transfer coupled with grey-body surface radiation for the numerical simulation of chemical vapor deposition processes

In this paper an advanced numerical approach for the simulation of epitaxial growth in metalorganic chemical vapor deposition reactors (MOCVD) is presented. The mathematical model is based on the conservation equations for momentum and heat transfer combined with mass transfer including thermodiffusion and chemical reactions. The thermal radiation analysis assumes a non-participating medium and semi-transparent quartz walls. The radiation heat transfer is coupled with convection and conduction. The heat conduction includes thermal solid/fluid interactions between the gas and solid parts of the reactor. The model is implemented in a finite volume numerical solution procedure on block-structured non-orthogonal grids for two-dimensional (plane and axisymmetric) laminar flows. To speed up the convergence of the computations, a “full approximation scheme” multigrid technique is employed. In order to demonstrate the ability of the present method to analyze complex problems, investigations for horizontal CVD reactor configurations are presented. These problems include very complicated radiative heat transfer, buoyancy-driven flow, combined heat transfer in the reactor walls and the susceptor, as well as the transport of chemically reacting components. The simulated temperature distribution is compared with well-known temperature measurements [1] [L. Stock and W. Richter, J. Crystal Growth 77 (1986) 144] and good agreement with them is achieved. The growth of GaAs from trimethylgallium (TMGa), arsine, and hydrogen was considered where the deposition process is assumed to be in the transport-limited regime. The predicted deposition rates in the reactor fairly well compare with the available experimental results from literature.

Numerical study of the turbulent flow over and in a model forest on a 2D hill

Numerical predictions are presented for flow through and in a model forest on a two-dimensional hill. Calculations are performed with a finite volume numerical algorithm, using non-staggered non-orthogonal grid systems. To eliminate false diffusion errors, the convective terms in the governing equations are discretized using a highly accurate numerical scheme constructed from the class of total variation diminishing (TVD) schemes. The time averaged Navier-Stokes equations are closed by an extended k- ϵ eddy viscosity model including terms due to the drag caused by the canopy. The predictions are compared with reported LDV measurements obtained in an atmospheric boundary layer wind tunnel for different forested hill geometrical configurations. The calculations show good agreement with measurements concerning the strong influence of the tree cover on the separated flow region.