Line Successive Overrelaxation Research Articles

Natural Convection Heat and Mass Transfer in the Vertical Cylindrical Porous Channel Under the Effects of Time-Periodic Boundary Condition

Abstract Heat and mass transfer are investigated numerically with steady-state laminar natural convection through a vertical cylindrical enclosure filled with a liquid-saturated porous medium. The vertical wall is under a constant magnetic field and various durations of periodic heating boundary condition; the top and bottom surfaces are kept at a constant cold temperature. Continuity, momentum, and energy equations are transformed to dimensionless equations. The finite difference approach with the line successive over-relaxation (LSOR) method is used to obtain the computational results. This study covers the heat transfer, the temperature distribution, and the velocity field in the domain under the variation of different parameters. The code used is validated by modifying it to analyze the Nusselt number in the existing experimental literature of Izadpanah et al. (1998, “Experimental and Theoretical Studies of Convective Heat Transfer in a Cylindrical Porous Medium,” Int. J. Heat Fluid Flow, 19(6), pp. 629–635). This work shows that Nusselt number decreases (with varying gradient) as the aspect ratio increases, and that it increases as the Rayleigh number increases. The centerline temperature has a proportional relationship with the heating amplitude and the heating period (as the system receives more heat) and is inversely proportional with Rayleigh number. Increasing the Rayleigh number causes increased convective velocity, which affects the position of the hot region, and causes a decrease in the temperature field. Increasing the aspect ratio results in a warm stream at the center of the cylinder, and when the time period of the heating increases, the circulation becomes faster and the intensity of the temperature contour layers decreases. In this work, a correlation for Nu as a function of the mentioned parameters is developed.

On the Line Successive Overrelaxation Method

A line version of the KSOR method is introduced, LKSOR method. Comparison of the performance of some different iterative techniques with their line format (Jacobi – Gauss Seidel and SOR) are considered. Implementation of LKSOR method for several different formulas in different mesh geometries is discussed. The proposed method considers the advantages of the LSOR in addition to those of the KSOR. A graphical representation of the behavior of the spectral radius near the optimum value illustrates the smoothness in the selection of relaxation parameters.

Grid-Generation Algorithms for Complex Glaze-Ice Shapes Reynolds-Averaged Navier–Stokes Simulations

The paper presents the developments of novel mesh generation algorithms over complex glaze-ice shapes containing multicurvature ice-accretion geometries, such as single/double ice horns. The twofold approaches tackle surface geometry discretization as well as field mesh generation. First, an adaptive curvilinear curvature control algorithm is constructed, solving a one-dimensional elliptic partial differential equation with periodic source terms. This method controls the arc length grid spacing, so that high convex and concave curvature regions around ice horns are appropriately captured, and is shown to effectively treat the grid shock problem. Second, a novel blended method is developed by defining combinations of source terms with two-dimensional elliptic equations. The source terms include two common control functions, Sorenson and Spekreijse, and an additional third source term to improve orthogonality. This blended method is shown to be very effective for improving grid quality metrics for complex glaze-ice meshes with Reynolds-averaged Navier–Stokes resolution. The performance in terms of residual reduction per nonlinear iteration of several solution algorithms (point–Jacobi, Gauss–Seidel, alternating direction implicit, point, and line Successive Over-Relaxation) are discussed within the context of a full multigrid operator. Details are given on the various formulations used in the linearization process. It is shown that this performance of the solution algorithm depends on the type of control function used. Finally, the algorithms are validated on standard complex experimental ice shapes, demonstrating the applicability of the methods.

Computational modeling and analysis of thermal characteristics of a rotating spacecraft subjected to solar radiation

Mohamed A. Gadalla and Essam WahbaCollege of Engineering, Mechanical Engineering DepartmentAmerican University of Sharjah, Sharjah 26666, U. A. E.This paper presents a computational modeling for the temperature distributionof a rotating thick-walled cylindrical spacecraft subjected to solar radiation. The innersurface of the spacecraft is thermally insulated while the outer surface is subjected toconcurrent events of solar incidence and radiative heat dissipation to space. Thegoverning equation for the normalized temperature is discretized using a finitedifference scheme and Successive Line Over-Relaxation (SLOR) is used to solve theresulting system of algebraic equations. Numerical simulations of temperature distri-bution on the spacecraft for different spinning speeds, angular positions, and differentradii are discussed and evaluated for both linearized and nonlinear boundary condi-tions. Comparative analysis between the computational modeling and the exactanalytical solution for the linearized boundary condition is presented. The resultsindicate that the outer surface temperature distribution of the spacecraft is nearlyindependent of the angular position; at sub-cylindrical surface, this independence isachieved at low angular velocity. Moreover, numerical simulations show that the useof the linearized boundary condition at the outer surface presents a good approximationfor the case of high-speed spinning spacecrafts, while it results in significant errors inthe temperature field in the case of stationary and low-speed spinning spacecrafts.© 2011 Wiley Periodicals, Inc. Heat Trans Asian Res, 40(7), 655–676, 2011; Publish-ed online 21 July 2011 in Wiley Online Library (wileyonlinelibrary.com/journal/htj).DOI 10.1002/htj.20368

Optimization studies of fuel loading pattern for a typical Pressurized Water Reactor (PWR) using particle swarm method

A three dimensional multi-energy group computer model PRISHA, which solves the neutron diffusion equations using finite difference method is developed for Pressurized Water Reactor (PWR). This computer code can find an optimum loading of a group of fresh fuel assemblies along with fuel assemblies of different exposures. The successive line over relaxation (SLOR) method is used to solve neutron diffusion equations. After validation of this part of computer code against an IAEA – PWR benchmark problem with 177 fuel assemblies in the core, particle swarm optimization (PSO) method is incorporated in the code for finding the optimum fuel loading pattern. A typical PWR core with 157 fuel assemblies, where 289 fuel pins are arranged in 17 × 17 rectangular arrays in a fuel assembly, was analyzed using this computer model for two cycles using PSO method. Different numbers of particles and iterations were used in PSO method. The results are found to be not very sensitive to either the number of particles or the number of iterations used in PSO method for considered case. However, a number of experiments have to be performed to arrive at the best global fitness parameter. Reasonably low power peaking factors were obtained for both the cycles.

Iterative methods for solving space-time one dimensional multigroup diffusion equations

The numerical methods of Successive Overrelaxation (SOR), Successive-Line Overrelaxation (SLOR) and Alternating Direction Implicit (ADI) are used in this research to solve space-time one-dimensional diffusion equations. The results of this research have been compared with the ANL-Benchmark Problem BSS-6-A2. In general, the results are in agreement with the Argonne National Laboratory Benchmark Source Situation (BSS-6-A2). However, there is very little difference between SOR and ANL-BSS-6-A2 because of inner iterations in the SOR method. This research shows that the SLOR and ADI methods are more accurate and more efficient than the SOR method in solving space-time diffusion equations. SLOR is the most efficient in the case of a big number of mesh elements; however, the ADI method is the most efficient in the case of a small number of mesh elements.

On the formulation of sea-ice models. Part 1: Effects of different solver implementations and parameterizations

This paper describes the sea ice component of the Massachusetts Institute of Technology general circulation model (MITgcm); it presents example Arctic and Antarctic results from a realistic, eddy-admitting, global ocean and sea ice configuration; and it compares B-grid and C-grid dynamic solvers and other numerical details of the parameterized dynamics and thermodynamics in a regional Arctic configuration. Ice mechanics follow a viscous-plastic rheology and the ice momentum equations are solved numerically using either line-successive-over-relaxation (LSOR) or elastic–viscous-plastic (EVP) dynamic models. Ice thermodynamics are represented using either a zero-heat-capacity formulation or a two-layer formulation that conserves enthalpy. The model includes prognostic variables for snow thickness and for sea ice salinity. The above sea ice model components were borrowed from current generation climate models but they were reformulated on an Arakawa C grid in order to match the MITgcm oceanic grid and they were modified in many ways to permit efficient and accurate automatic differentiation. Both stress tensor divergence and advective terms are discretized with the finite-volume method. The choice of the dynamic solver has a considerable effect on the solution; this effect can be larger than, for example, the choice of lateral boundary conditions, of ice rheology, and of ice-ocean stress coupling. The solutions obtained with different dynamic solvers typically differ by a few cm s −1 in ice drift speeds, 50 cm in ice thickness, and order 200 km 3 yr −1 in freshwater (ice and snow) export out of the Arctic.

Automatic elliptic grid generation by an approximate factorisation

A procedure for automatic numerical generation of a structured grid system with coordinate lines coinciding with all boundaries of a general two-dimensional region containing a body of arbitrary shape is presented. The solution procedure incorporated the method of false transients and the approximate factorisation algorithm, where a sequence of time steps is cycled in a geometric fashion with repeated endpoints, and has a capability for clustering grid lines close to the body. The procedure requires significantly much less computational effort to obtain a converged solution than a point or line successive over-relaxation iterative scheme. Although, the superiority of the presented algorithm has been demonstrated for the grid generation problem, it can be utilised for other problems requiring the solution of a set of elliptic partial differential equations of similar nature.

Effect of magnetic field on natural convection in a vertical cylindrical annulus

Natural convection of a low Prandtl number electrically conducting fluid ( Pr = 0.054) under the influence of either axial or radial magnetic field in a vertical cylindrical annulus has been numerically studied. The inner and outer cylinders are maintained at uniform temperatures and the horizontal top and bottom walls are thermally insulated. A finite difference scheme consisting of alternating direction implicit (ADI) method and successive line over relaxation (SLOR) method is used to solve the vorticity stream function formulation of the problem. Detailed numerical results of heat transfer rate, temperature and velocity fields have been presented for 1 ⩽ λ ⩽ 10, 0.5 ⩽ A ⩽ 2, 10 3 ⩽ Ra ⩽ 10 6 and 0 ⩽ Ha r , Ha x ⩽ 10 2. The computational results reveal that in shallow cavities the flow and heat transfer are suppressed more effectively by an axial magnetic field, whereas in tall cavities a radial magnetic field is more effective. It is also found that the flow oscillations can be suppressed effectively by imposing an external magnetic field. The average Nusselt number increases with radii ratio but decreases with the Hartmann number. Further, the present numerical results are shown to be in good agreement with the available benchmark solutions under the limiting conditions.

A new simulation framework for predicting the onset and effects of fines mobilization

Fines release and migration is a universal problem in the production of oil from poorly consolidated sandstone reservoirs. This problem can result in the changes of porosity and permeability. It may not only damage a production facility, but it can also have a profound effect on oil recovery, resulting from the change in heterogeneity of the oil formation. Based on the macroscopic continuous porous media, continuity equations for multiphase flow in oil formations, and the theories of fines release and migration, a three-dimensional (3D) field scale mathematical model describing migration of fines in porous media is developed. The model is solved by a finite-difference method and the line successive over relaxation (LSOR) technique. A numerical simulator is written in Fortran 90 and it can be used to predict (1) the ratio of fines to production liquid volume, (2) the permeability change caused by colloidal and hydrodynamic forces resulting from fines release and migration, and (3) production performance. The numerical results of the one-dimensional model were verified by the data obtained by core displacement experiments. The sensitivity of numerical results with grid block size was studied by coarse grids, moderate grids, and fine grids. In addition, an oil field example with five-spot patterns was made on the numerical simulator. The results show that fines migration in an oil formation can accelerate the development of heterogeneity of the reservoir rock, and has an obvious influence on production performance, i.e., water drive front, water-cut trends, and oil recovery.

Numerical Study of Natural Convection in a Vertical Cylindrical Annulus Using a Non-Darcy Equation

A numerical investigation of transient free convection in a vertical cylindrical annulus filled with a fluid saturated porous medium with the inner wall heated to a uniform temperature, the outer wall cooled to a uniform temperature and maintaining top and bottom boundaries at adiabatic condition, has been carried out. A finite difference implicit method which incorporates upwind differencing for nonlinear convective terms and the successive line over relaxation (SLOR) method for convergence are used to solve the coupled nonlinear governing equations. Heat transfer rate and flow fields are obtained for wide range of Rayleigh numbers Ra ranging from 103 to 106, Darcy numbers Da ranging from 10-1 to 10-3, radii ratio λ. in the range 1 ≤ λ ≤10, and aspect ratio A in the range 1 ≤ A ≤ 2. The numerical results indicate that the temperature and velocity fields are significantly modified and the heat transfer rate is decreased as the Darcy number decreases. The temperature and velocity fields are significantly modified by the radii ratio. The rate of heat transfer increases with an increase in radii ratio. At a high Rayleigh number the curvature effect on heat transfer is insignificant. The numerical results farther indicate that an increase in viscosity ratio (Λ) reduces the Nusselt number by enhancing the contribution of the viscous diffusion term. Copyright © 2002 by Begell House, Inc.

Fourth‐order exponential finite difference methods for boundary value problems of convective diffusion type

AbstractMethods based on exponential finite difference approximations of h4 accuracy are developed to solve one and two‐dimensional convection–diffusion type differential equations with constant and variable convection coefficients. In the one‐dimensional case, the numerical scheme developed uses three points. For the two‐dimensional case, even though nine points are used, the successive line overrelaxation approach with alternating direction implicit procedure enables us to deal with tri‐diagonal systems. The methods are applied on a number of linear and non‐linear problems, mostly with large first derivative terms, in particular, fluid flow problems with boundary layers. Better accuracy is obtained in all the problems, compared with the available results in the literature. Application of an exponential scheme with a non‐uniform mesh is also illustrated. The h4 accuracy of the schemes is also computationally demonstrated. Copyright © 2001 John Wiley & Sons, Ltd.

On an efficient numerical method for modeling sea ice dynamics

A computationally efficient numerical method for the solution of nonlinear sea ice dynamics models employing viscous‐plastic rheologies is presented. The method is based on a semi‐implicit decoupling of the x and y ice momentum equations into a form having better convergence properties than the coupled equations. While this decoupled form also speeds up solutions employing point relaxation methods, a line successive overrelaxation technique combined with a tridiagonal matrix solver procedure was found to converge particularly rapidly. The procedure is also applicable to the ice dynamics equations in orthogonal curvilinear coordinates which are given in explicit form for the special case of spherical coordinates.

Efficient Implementation Of Local Grid Refinement Ina Generalized Compositonal Model

Abstract An efficient implementation of local grid refinement (LGR) in a generalized compositional model js presented, Highlights include:implementation for both block-centred and point-distributed grids,a highly efficient linear solver, with a novel multi-colour ordering, for both five-point and nine-point computational molecules, andefficient data structure which is essential for optimal vector and scalar performance, to maximize the benefits of LGR. For block-centred grids, refinement is performed by subdividing coarse gridblocks. For point-distributed grids, however, refinement is performed by subdividing along coarse grid lines. This gives rise to several additional complexities at the interfaces between different grids. The actual implementation is discussed. The linear solver differs from other methods (e.g., BEPS) in that it operates on the composite grid system. For a single grid system (no LGR), the solver uses redlblack ordering for five point molecules and four colour ordering for nine-point molecules. For a composite grid system, a novel repeated multi-colour ordering is employed. This solver- has good convergence properties due to better coupling between.subgrids, as compared to domain decomposition methods. In fact, convergence rates were found to be in line with those for single grids. The result is a linear solver which avoids the multiple linear solutions that domain decomposition methods entail. Further, the same solver can be used efficiently for both single grid and composite grid systems, eliminating the need to maintain extra solvers. Finally, simulations showing application to water and miscible floods are presented. They demonstrate the significant savings in CPU expense achieved with this model, while obtaining results of good accuracy. Introduction In recent years, local grid refinement techniques have become increasingly popular(l-8). They have been driven largely by the need to better characterize reservoir processes, which occur on many different scales. Modelling of near well behaviour (e.g., liquid dropout in gas condensate, coning), pilot studies, and geostatistically derived reservoir characterizations are some examples that would benefit greatly from local grid refinement. As an example, pilot studies have hitherto employed grids that are globally refined. This can be computationally expensive, and some times compromises must be made with regard to the boundary conditions (modelling of the interaction with the surrounding reservoir). Local grid refinement can generate accurate simulation results at significantly reduced computational expense. Local grid refinement, however, brings with it its own special problems, such as the calculation of intergrid fluid transfer and the solution of a pressure matrix with non-standard connections due to the multiple grids used. In fact, most of the literature on local grid refinement is devoted to the solution of these problems(4–7). This paper deals with the implementation of the local grid refinement feature for Cartesian grids in Amoco's Generalized Compositional Model (GCOMP)(9). We discuss technical questions such as generation of the composite grid pressure coefficients and their solution with a novel multiple grid, multi-colour z-line successive line over relaxation (SLOR) linear solver, and we present results of some simulation studies using the local grid refinement feature.

Combined surface tension and buoyancy-driven convection in a rectangular open cavity in the presence of a magnetic field

A numerical study is conducted to understand the effect of magnetic field on the flow driven by the combined mechanism of buoyancy and thermocapillarity in a rectangular open cavity filled with a low Prandtl number fluid ( Pr = 0.054). The two side walls are maintained at uniform but different temperatures θ h and θ c ( θ h > θ c ), while the horizontal top and bottom walls are adiabatic. A finite difference scheme consisting of the ADI (Alternating Direction Implicit) method, which incorporates upwind differencing for non-linear convective terms and the SLOR (Successive Line Over Relaxation) method are used to solve the coupled non-linear governing equations. Computations are carried out for a wide range of Grashof number Gr ranging from 2 × 10 4 to 2 × 10 6, Marangoni number Ma from 0 to 10 5 and Hartmann number Ha from 0 to 100. The detailed flow structure and the associated heat transfer characteristics inside the cavity are presented. At large Ma, two counter-rotating cells are formed at the upper half and lower half of the enclosure. As Ha increases, the temperature field resembles that of a conduction type and the streamlines are elongated in nature in the horizontal direction. The upper cell is crowded and stretched along the free surface. The average Nusselt number increases with Ma but decreases with Ha.

Effect of a magnetic field on free convection in a rectangular enclosure

Natural convection of an electrically conducting fluid in a rectangular enclosure in the presence of a magnetic field is studied numerically. The two vertical side walls are held isothermally at temperatures θ h and θ c, while the horizontal top and bottom walls are adiabatic. A finite difference scheme consisting of modified ADI (Alternating Direction Implicit) method and SLOR (Successive Line Over Relaxation) method is used to solve the vorticity-stream function formulation of the problem. Numerical predictions are obtained for a wide range of Grashof number Gr and Hartmann number Ha at the Prandtl number Pr = 0.733. At low Grashof number regime with weak magnetic field, a circulating flow is formed in the enclosure. When the magnetic field is relatively strengthened, the thermal field resembles that of a conductive distribution, and the fluid in much of the interior is nearly stagnant. Further, when the magnetic field is weak and the Grashof number is high, the convection is dominant and vertical temperature stratification is predominant in the core region. However, for sufficiently large Ha, the convection is suppressed and the temperature stratification in the core region diminishes. The numerical results show that the effect of the magnetic field is to decrease the rate of convective heat transfer. The average Nusselt number decreases with an increase of Hartmann number.

Transonic Euler computation in streamfunction co‐ordinates

AbstractA new approach has been developed to calculate two‐dimensional steady transonic flows past aerofoils using the Euler equations in streamfunction co‐ordinates. Most existing transonic computation codes require the use of a grid generator to determine a suitable distribution of grid points. Although simple in concept, the grid generation may take a considerable proportion of the CPU time and storage requirements. However, this grid generation step can be avoided by introducing the von Mises transformation, which produces a formulation in streamwise and natural body‐fitting co‐ordinates. In this work a set of Euler equivalent equations in streamfunction co‐ordinates is formulated, consisting of three equations with three unknowns; one is a geometric variable, the streamline ordinate y, and the other two are physical quantities, the density ρ and the vorticity ω. To solve these equations, typedependent differencing, development of a shock point operator, marching from a non‐characteristic boundary and successive line overrelaxation are applied. Particular attention has been paid to the supercritical case where a careful treatment of the shock is essential. It is shown that the shock point operator is crucial to accurately capture shock waves. The computed results show excellent agreement with existing experimental data and other computations.

A hybrid integral equation finite volume scheme for transonic potential flow about complex configurations

AbstractA hybrid integral equation finite volume scheme has been developed for the calculation of transonic potential flow about complex configurations. A new technique has been used for evaluating the potential values in the field cells intersecting the body surface panels. These potential values then serve as Dirichlet boundary conditions for computing the potentials in the field by a finite volume Successive Line Over Relaxation (SLOR) scheme. In this approach there is no need to evaluate the potentials anywhere in the field by direct application of Green's third identity, thus significantly reducing computer processing time and storage requirement, while improving accuracy of surface pressure prediction and shock capture, as results indicate. The capability of tackling additional complex geometry with ease, the primary advantage of the integral equation approach, is demonstrated by using the same field grid for wing-alone and wing-body combination cases, while maintaining the solution accuracy.

Three-dimensional initial data for the collision of two black holes.

We describe three numerical approaches to the construction of three-dimensional initial data for the collision of two black holes. The first of our approaches involves finite differencing the 3 + 1 Hamiltonian constraint equation on a \ifmmode \check{C}\else \v{C}\fi{}ade\ifmmode \check{z}\else \v{z}\fi{} coordinate grid. The difference equations are then solved via the multigrid algorithm. The second approach also uses finite-difference techniques, but this time on a regular Cartesian coordinate grid. A Cartesian grid has the advantage of having no coordinate singularities. However, constant coordinate lines are not coincident with the throats of the black holes and, therefore, special treatment of the difference equations at these boundaries is required. The resulting equations are solved using a variant of line-successive overrelaxation. The third and final approach we use is a global, spectral-like method known as the multiquadric approximation scheme. In this case functions are approximated by a finite sum of weighted quadric basis functions which are continuously differentiable. We discuss particular advantages and disadvantages of each method and compare their performances on a set of test problems.

A Finite-Difference Scheme for Two Dimensional Incompressible Turbulent Flows Using Curvilinear Coordinates.

An implicit time-marching finite-difference method, which has been developed previously to solve the incompressible Navier-Stokes equations of contravariant velocities in curvilinear coordinates, is extended to a turbulent flow scheme employing a suitable low Reynolds number κ-e model. The present method has second-order accuracy in time with application of the Crank-Nicholson scheme, and utilizes the QUICK (Quadratic Upstream Interpolation for Convective Kinematics) upwind-difference scheme in space. The elliptic equation of pressure is solved by means of the Tschebyscheff SLOR (Successive Line Over Relaxation) method with alteration of the computational directions. The present implicit turbulent flow scheme is stable under the proper boundary conditions, since spurious error and numerical instabilities can be suppressed by employing a staggered grid. Numerical calculations are performed for turbulent flows through a two-dimensional cascade. Computed results of the cascade turbulent boundary layers and the wake profiles are shown. The calculated surface pressure distribution is in good agreement with the experimental data.