Non-orthogonal Grids Research Articles

A Methodology to Solve 2D and Axisymmetric Radiative Transfer Problems Using a General 3D Solver

A method to solve the radiative transfer equation (RTE) for absorbing-emitting and/or scattering media for 2-D and axisymmetric geometries using a general 3-D solver with a special treatment of the boundary condition in the third direction is presented. It allows a choice of first- or second- order schemes and can be used with non-orthogonal hexahedral grids for complex domains. Two-dimensional or axisymmetric problems are treated as different special cases of a three-dimensional problem. The method is tested on axisymmetric problems with absorbing-emitting and/or scattering media and on a 2D planar problem with a transparent medium and validated by comparisons with benchmark solutions.

Nonorthogonal LOD-FDTD Method for EM Scattering from Two-Dimensional Structures

In this paper, a nonorthogonal locally 1-D finite difference time-domain (LOD-NFDTD) method is presented for electromagnetic (EM) scattering involving curvilinear coordinate system. Formulations of scattered field and convolutional perfectly matched layer absorbing boundary condition in generalized nonorthogonal grids for LOD-NFDTD are also presented. The nonorthogonal grids are used to fully mesh the computational domain which leads to efficient computation. Moreover, the proposed technique requires fewer arithmetic operations than the alternating direction implicit NFDTD method leading to a reduction of CPU time. The numerical dispersion of the proposed method as a function of Courant-friedrich-lewy number (CFLN) is also presented. Computational results for EM scattering from 2-D conducting, dielectric, and coated cylinders as well as overfilled dielectric and bent cavity structures are presented.

Numerical Investigation of Heat Transfer and Fluid Flow Characteristics inside Wavy Channels Fully Filled with Porous Media

The combined effect of waviness and porous media on the convection heat transfer and fluid flow characteristics is numerically investigated. Two models of wavy walled channel fully filled with homogenous porous material are assumed. The first was the symmetric converging-diverging channel (case A), and the second was the channel with concave-convex walls (case B). The governing equations have been solved on non-orthogonal grid, which is generated by Poisson elliptic equations, based on ADI method. Nusselt number values are used to indicate whether any cases of corrugation studied may have led to an increase in the rate of heat transferred compared with the planar surface channel which is the purpose of the study. The results show that case A gives more enhancement in heat transfer than case B. However, the thermal performance of the wavy channels (cases A & B) is better than the straight channel (simple duct).

Nodal Integral Method Using Quadrilateral Elements for Transport Equations: Part 2-Navier-Stokes Equations

A numerical scheme for the Navier-Stokes equations in an irregular shaped domain using the nodal integral method (NIM) is developed. A mapping similar to the convection-diffusion article (Part 1) is used for the Navier-Stokes equations using the NIM in an arbitrary-shaped domain. Use of a recently developed pressure correction-based iterative scheme for the NIM ensures that the final solution satisfies the continuity condition. Lid-driven and buoyancy-driven flows in a skewed cavity are used as two test cases for comparison and verification. From the detailed comparative study of the results obtained by the NIM and very-fine-grid results (using finite-volume or similar approaches) of previous studies, it is established that this NIM-based scheme for the Navier-Stokes equation retains its capability to produce accurate results in comparatively much coarser grids, even with nonorthogonal grids

The Radiative Boundary Design of a Hexagonal Furnace Filled With Gray and Nongray Participating Gases

The inverse radiative boundary design problem in a hexagonal furnace is numerically investigated. The aim of this study is to find the strength of heaters in a two-dimensional (2D) enclosure to produce the desired temperature and heat flux distribution on the design surface. Conjugate gradients method is chosen to perform the iterative search procedure for obtaining the optimal solution. The medium is considered participating and both gray and nongray cases are studied. FTn finite volume method (FVM) with nonorthogonal grid is employed to solve the radiative transfer equation in the furnace. In nongray cases, the medium is filled with combustive gas products. To predict nongray behavior properly, the SLW model is used. In problems with gray medium, the effects of wall emissivity, absorption coefficient, and scattering albedo are studied. For nongray problems, the effects of gas concentration and temperature of single species and gas mixtures are analyzed. In all cases, heater power can be estimated precisely and as the results show, estimated heat flux on the design surface is very close to the desired value. For this reason, the maximum root mean square error is less than 1.6%.

A co-volume scheme for the rotating shallow water equations on conforming non-orthogonal grids

A co-volume scheme is introduced for the rotating shallow water equations, in which both velocity components are specified on cell edges, and the thickness variables evolve on both the primary and the dual cell centers. The scheme applies to generic, conforming and non-orthogonal staggered grids, including the widely used lat–lon quadrilateral grids and the Delaunay–Voronoi tessellations. It can be viewed either as coupled C-grid schemes on the primary and dual meshes, or as an generalization of the traditional E-grid scheme on a new non-overlapping grid. Linear dispersive wave analysis shows that, the dispersive relations resolved by either the primary or the dual mesh of a uniform quadrilateral staggered grid is the same as those of the Z-grid scheme. The total wavenumber space resolved by the staggered grid is twice as large, on which the co-volume behaves exactly like the E-grid scheme. On a uniform hexagon-triangular staggered grid, the co-volume has two steady modes and two inertial-gravity modes on the hexagonal mesh, and one steady mode, two inertial gravity modes, and two spurious modes on the triangular mesh. On the wavenumber space resolved by either the hexagonal or the triangular mesh, the inertial-gravity wave modes remain positive and largely monotone. For the nonlinear shallow water equations, the co-volume scheme is shown to preserve the potential vorticity dynamics and the total energy exactly. Numerical results are presented to corroborate and supplement the analyses.

A three-dimensional semi-implicit unstructured grid finite volume ocean model

A new three-dimensional semi-implicit finite-volume ocean model has been developed for simulating the coastal ocean circulation, which is based on the staggered C-unstructured non-orthogonal grid in the horizontal direction and z -level grid in the vertical direction. The three-dimensional model is discretized by the semi-implicit finite-volumemethod, in that the free-surface and the vertical diffusion are semi-implicit, thereby removing stability limitations associated with the surface gravity wave and vertical diffusion terms. The remaining terms in the momentum equations are discretized explicitly by an integral method. The partial cell method is used for resolving topography, which enables the model to better represent irregular topography. The model has been tested against analytical cases for wind and tidal oscillation circulation, and is applied to simulating the tidal flow in the Bohai Sea. The results are in good agreement both with the analytical solutions andmeasurement results.

三川合流部における洪水流と河床変動解析―格子非直交成分の運動量フラックスの導入

In this study, the quasi-3D model composed of depth-integral vortex equation and shallow water equations was extended on the basis of finite volume method employing non-orthogonal grids for properly evaluating complex boundary of the confluences. And, we developed the flux integral method considering non-orthogonal direction of grid line based on the CIP-CSL scheme for evaluating the convection terms more accurately also in the complex flow field of the confluence. The extended quasi-3D model was applied to recent large floods at the Go River confluences to investigate the flood flow and bed variation during the flood events. From the calculation results, we clarified that separation zones formed at the confluences have pronounced effects on the three dimensional flow structure and bed variations for the downstream channel.

NUMERICAL SIMULATION OF FLOW AND LOCAL SCOUR AT TWO SUBMERGED-EMERGENT TANDEM CYLINDRICAL PIERS

In this paper, the flow and local scour variation around two submerged and un-submerged tandem piers are studied using 3D flow model where the upstream pier is submerged while the downstream pier is emergent. The model uses a finite-volume method to solve the non-transi ent Navier-Stokes equations for three dimensions on a general non-orthogonal grid. The   k turbulence model is used to solve the Reynolds-stress term. The numerical model solves the sediment continuity equation in conjunction with van-Rijn’s bed-load sediment transport formula to simulate the bed evolution. The 3D flow model is verified through experimental study in a non cohesive bed material in an experimental flume. The different causes of local scour around two submerged and unsubmerged piers are simulated well, such as bow flow, down flow, horseshoe vortex, pressure variation and lee-wake vortex. It is found from this study that the maximum local scour depth by interaction between two tandem submerged unsubmerged piers depends on submersion ratio of upstream pier, the densimetric Froude number, the longitudinal distance between piers and the ratio of pier diameter to channel bed width. The maximum scour depth decreases by increasing the submerged pier height then begin to increase by increasing the submerged pier to a height larger than half the water depth and in general the maximum scour depth is less than that of two unsubmerged piers. The results show good agreement between simulation and experimental results. Also, empirical equations are developed for computing the maximum scour depth due to the interaction between two submerged unsubmerged piers with circular shapes as a function of submergence ratio, piers spacing, densimetric Froude number and channel width to pier diameter ratio.

Non-Orthogonal Grids in Two-Dimensional Transmission-Line Metamaterials

Characteristics of two-dimensional metamaterials based on non-orthogonal grids of transmission lines are investigated. From periodic analysis on a unit cell, the dispersion function is derived and compared with the orthogonal grid case. Isomorphism between Kirchhoff's circuital laws and two-dimensional Maxwell's equations is utilized to characterize the effective medium parameters. It is shown that a non-orthogonal grid structure is effective in realizing two-dimensional full-tensor anisotropic medium parameters in a simple geometrical configuration. For verification, a design example is provided where a probe current radiates near a planar interface between two anisotropic media and the results are compared with the analytical solution.

Investigation of Instabilities Arising with Non-Orthogonal Meshes Used in Cell Centred Elliptic Finite Volume Computations

Transport processes in most engineering applications occur within complex geometries. In engineering practice, users rely heavily on commercial mesh generators, which can produce unacceptably skewed meshes. Convergence behaviour and absolute accuracy in finite volume CFD computations depend critically on mesh quality and in particular, mesh orthogonality. In this paper, the effects of non-orthogonality on the main component algorithms of pressure-correction type cell-centred finite volume codes are closely examined, systematically adjusted and tested. The modifications to the pressure correction method applied to cases using non-orthogonal grids are described. The SIMPLEC algorithm [1], with the aid of an inverse square distance interpolation, is used for overcoming instabilities arising in a few problematic cells. Solution instabilities which arise when using hexahedral or tetrahedral meshes are attenuated by bounding the maximum and minimum values of solved variables within a physically realistic range. The consistency and accuracy of the proposed method are compared with benchmark solutions [2] available in the literature. The usefulness of the present method is demonstrated by its application to illustrative problems for which comparison data are available.

Aluminum alloy profile extrusion simulation using finite volume method on nonorthogonal structured grids

Purpose – The paper aims to use the finite volume method widely used in computational fluid dynamics to avoid the serious remeshing and mesh distortion during aluminium profile extrusion processes simulation when using the finite element method. Block-structured grids are used to fit the complex domain of the extrusion. A finite volume method (FVM) model for aluminium extrusion numerical simulation using non-orthogonal structured grids was established. Design/methodology/approach – The influences of the elements ' nonorthogonality on the governing equations discretization of the metal flow in aluminium extrusion processes were fully considered to ensure the simulation accuracy. Volume-of-fluid (VOF) scheme was used to catch the free surface of the unsteady flow. Rigid slip boundary condition was applied on non-orthogonal grids. Findings – This paper involved a simulation of a typical aluminium extrusion process by the FVM scheme. By comparing the simulation by the FVM model established in this paper with ...

A Finite-Volume Model for Fluid Flow and Nonequilibrium Heat Transfer in Conjugate Fluid-Porous Domains Using General Unstructured Grids

A numerical model for computing fluid flow and heat transfer, under the assumption of local thermal nonequilibrium, is proposed for use with general unstructured, nonorthogonal grids. This model introduces novel interface conditions which are physically-reasoned and ensure strong coupling between the pressure and velocity fields. Special attention is given to the numerical approximation of diffusive and advective fluxes, pressure forces in the momentum equations, pressure–velocity coupling, and gradient reconstruction at interfaces while maintaining second-order accuracy. The resulting model is very robust and is shown to produce physically reasonable results for high laminar Reynolds numbers on nonorthogonal grids.

Unified Curvilinear Ocean Atmosphere Model (UCOAM): A vertical velocity case study

In this paper, the Unified Curvilinear Ocean Atmosphere Model (UCOAM) is described. UCOAM is a non-hydrostatic LES model, capable of reading curvilinear orthogonal and non-orthogonal grid in all the three dimensions including the vertical. Furthermore, the interaction of bottom bathymetry with the flow field and particularly, the vertical component of the velocity in a fully curvilinear grid and a sigma grid are presented.

Numerical simulation of 2D lid-driven cavity flow with CLEARER algorithm on extremely highly skewed grids at high Reynolds numbers

It is crucial to deal with the grid non-orthogonality effectively in solving the flow in complex geometries, especially at high Reynolds numbers. In this study, the newly proposed Coupled and Linked Equations Algorithm Revised-ER (CLEARER) algorithm is adopted to solve this problem successfully. In CLEARER algorithm the second relaxation factor is introduced in constructing the contravariant interface velocities, by setting it to a low value. CLEARER algorithm can overcome the severe grid non-orthogonality and non-linearity of equations effectively. After the numerical results with CLEARER are validated with the benchmark solutions, this algorithm is used to solve the lid-driven flow in inclined cavity with inclination angles varying from 10 to 170∘, and Reynolds numbers varying from 5000 to 15 000. The streamlines and the centerline velocity distributions are provided in detail for all cases, which may offer some guidance for the study in this area. Copyright © 2010 John Wiley & Sons, Ltd.

Modeling of Aluminum Extrusion Process Using Non-Orthogonal Block Structured Grids Based FVM

The mathematic model of 3D aluminum extrusion processes using finite volume method (FVM) was established in this paper. The basic theories and rigid-plastic flow theories of this model were researched and built. Non-orthogonal structured grids were used to match complex geometric boundaries and local refinement of grids was also realized. The collocated arrangement is used to discretize the governing equations on non-orthogonal grids directly, pressure oscillations bring by this arrangement and error caused by grid’s non-orthogonality is eliminated by special treatment. A pocket die extrusion process was simulated using the program developed in this paper. The simulation results were also compared with that simulated by FEM software Deform in the same process, material and die conditions. The feasibility and efficiency of the mathematic model built in this paper was demonstrated by the simulation results and the comparison.

Numerical simulation of 3D unsteady aluminium extrusion process using finite volume method based on non-orthogonal block structured grids

The mathematic model of three-dimensional aluminium profile extrusion processes using the finite volume method was established in the present paper. Basic theories and key technologies of the Simple method using the collocated arrangement strategy based on boundary fitted non-orthogonal block structured grids were studied. The governing equations were directly discretised on boundary fitted non-orthogonal grids. The coordinate transformation between the Cartesian rectangular coordinateand the curvilinear coordinate was avoided. The negative effects of grids' non-orthogonality, such as the dislocation between the lines connecting the centre points of adjacent elements and the centre of their interface, were compensated. The volume of fluid (VOF) scheme was used to capture the free surface of the deforming material. A program was written according to the above theories and equations. A typical three-dimensional aluminium profile extrusion process was simulated by use of the program developed in the present paper. The simulation results were also compared with that simulated by Deform-3D in the same process, material and die conditions. The feasibility of the mathematic model built in the present paper was demonstrated by the simulation results and the comparison.

Modeling of the 3D Unsteady Melt Flow in an Industrial-Scale Cz-Si Crystal Growth Using LES Method

We present a computational model for the 3D unsteady turbulent melt flow in an industrial-scale Cz-Si growth basing on the Large Eddy Simulation (LES) method in curvilinear grids. The filtered governing equations were solved basing on the finite-volume method for non-orthogonal grids. The turbulent transport of momentum and heat in the melt was investigated. The first and the second order statistics quantities of velocity and temperature fields were computed for analysis of the turbulent fluctuation in the melt flow. The effects of crystal and crucible rotation, thermal buoyancy and thermo-capillary on the melt flow were studied comprehensively through analysis of the instantaneous and mean velocity fields. The turbulent nature consisted of the unsteady and three-dimensional behavior of the melt flow was verified. Temperature fluctuations are obviously observed close to the crystallization zone.

Three‐dimensional finite difference saturated‐unsaturated flow modeling with nonorthogonal grids using a coordinate transformation method

Study of the saturated‐unsaturated flow in porous media is of interest in many branches of science and engineering. Among the various numerical simulation methods available, the finite difference method is advantageous because it offers simplicity of discretization. This method has been widely used for simulating saturated‐unsaturated flows. However, the simulation of geometrically complex flow domains requires the use of high‐resolution grids in conventional finite difference models because conventional finite difference discretization assumes an orthogonal coordinate system. This makes a finite difference model computationally less efficient than other numerical models that can treat nonorthogonal grids, such as the finite element model and finite volume model. To overcome this disadvantage, we use a coordinate transformation method and develop a multidimensional finite difference model for simulating saturated‐unsaturated flows; this model can treat nonorthogonal grids. The cross‐derivative terms derived by the coordinate transformation method are evaluated explicitly for ease of coding. Therefore, a 7 point stencil is used for implicit terms in the iterative calculation, as in the case of conventional finite difference models with an orthogonal grid. We assess the performance of the proposed model by carrying out test simulations. We then compare the simulation results with dense grid solutions in order to evaluate the numerical accuracy of the proposed model. To examine the performance of the proposed model, we draw a comparison between the simulation results obtained using the proposed model and the results obtained by using (1) a model in which all terms are considered fully implicitly, (2) a finite element model, and (3) a conventional finite difference model with a high‐resolution orthogonal grid.

Modeling of Hydrological Processes Using Unstructured and Irregular Grids: 2D Groundwater Application

To better handle landscape heterogeneities in distributed hydrological modeling, an earlier work proposed a discretization based on nested levels, which leads to fully unstructured modeling meshes. Upon such a discretization, traditional numerical solutions must be adapted, especially to describe lateral flow between the unstructured mesh elements. In this paper, we illustrated the feasibility of the numeric solution of the diffusion equation, representing groundwater flow, using unstructured meshes. Thus, a two-dimensional (2D) groundwater model (BOUSS2D), adapted to convex unstructured and irregular meshes was developed. It is based on the approximation of the 2D Boussinesq equation using numeric techniques suitable for nonorthogonal grids. The handling of vertical and horizontal aquifer heterogeneities is also addressed. The fluxes through the interfaces among joined mesh elements are estimated by the finite volume method and the gradient approximation method. Comparisons between the BOUSS2D predictions and analytical solutions or predictions from existing codes suggest the acceptable performance of the BOUSS2D model. These results therefore encourage the further development of hydrological models using unstructured meshes that are capable of better representing the landscape heterogeneities.