Different Discretization Schemes for Unstructured Grids
Similar to structured grids, there are two basic finite-volume techniques for the discretization of governing equations on unstructured grids. One is the cell-centered scheme, and another is the cell-vertex scheme.
For triangular grids, the shapes and the number of the control volumes for these two schemes are both different, e.g. for a cell-centered scheme, the control volumes are triangles while they are hexagons or polygons for a cell-vertex scheme, and the number of control volumes for a cell-centered scheme is larger than that for a cell vertex scheme. So our research focuses on the differences in accuracy and convergence rates between these two schemes.
Similar to structured grids, there are two basic finite-volume techniques for the discretization of governing equations on unstructured grids. One is the cell-centered scheme, and another is the cell-vertex scheme.
For triangular grids, the shapes and the number of the control volumes for these two schemes are both different, e.g. for a cell-centered scheme, the control volumes are triangles while they are hexagons or polygons for a cell-vertex scheme, and the number of control volumes for a cell-centered scheme is larger than that for a cell vertex scheme. So our research focuses on the differences in accuracy and convergence rates between these two schemes.
![图片](/uploads/2/5/4/3/25435923/5777870.png?437)
Numerical Example
One numerical example is a lid-driven cavity problem, and the computational domain and boundary conditions are illustrated in the first figure on the right. Re is set to be 1000.
The governing equation is discretized with finite volume method based on the cell-centered scheme and the cell-vertex scheme on triangular grids respectively.
The control volumes for the cell-centered scheme are triangles shown in figure (a), and for the cell-vertex scheme they are polygons shown in figure (b).
One numerical example is a lid-driven cavity problem, and the computational domain and boundary conditions are illustrated in the first figure on the right. Re is set to be 1000.
The governing equation is discretized with finite volume method based on the cell-centered scheme and the cell-vertex scheme on triangular grids respectively.
The control volumes for the cell-centered scheme are triangles shown in figure (a), and for the cell-vertex scheme they are polygons shown in figure (b).
Comparison
1. Accuracy
In order to compare the numerical errors of the cell-centered scheme and the cell-vertex scheme quantitatively, we calculated the U-velocity component and V-velocity component on a 160×160 mesh, and the results are set as the benchmark solution. The average absolute errors and relative errors can be obtained by comparing the numerical results with this benchmark solution.
Figure (a) shows comparison of velocity components on the mesh consisting of 1114 grids. Before grid independent solution is obtained, the solutions of the cell-vertex scheme are closer to the benchmark solution, which indicates that for a triangular mesh, the cell-vertex scheme is more accurate than the cell centered scheme, which makes it possible to use fewer grids to achieve the same accuracy and thus to save the computation time.
2. Convergence Rate
Under the identical circumstance including discretization of the governing equations, solution algorithm, iteration method for the algebraic equations, convergence criterion and mesh quality, the convergence rates between the two different schemes are compared. It can be seen in figure (b) that much less computation time is consumed by the cell-vertex scheme than by the cell-centered scheme. The reasons for the difference in convergence rates can be analyzed from the following two aspects.
1. Accuracy
In order to compare the numerical errors of the cell-centered scheme and the cell-vertex scheme quantitatively, we calculated the U-velocity component and V-velocity component on a 160×160 mesh, and the results are set as the benchmark solution. The average absolute errors and relative errors can be obtained by comparing the numerical results with this benchmark solution.
Figure (a) shows comparison of velocity components on the mesh consisting of 1114 grids. Before grid independent solution is obtained, the solutions of the cell-vertex scheme are closer to the benchmark solution, which indicates that for a triangular mesh, the cell-vertex scheme is more accurate than the cell centered scheme, which makes it possible to use fewer grids to achieve the same accuracy and thus to save the computation time.
2. Convergence Rate
Under the identical circumstance including discretization of the governing equations, solution algorithm, iteration method for the algebraic equations, convergence criterion and mesh quality, the convergence rates between the two different schemes are compared. It can be seen in figure (b) that much less computation time is consumed by the cell-vertex scheme than by the cell-centered scheme. The reasons for the difference in convergence rates can be analyzed from the following two aspects.
- The number of control volumes for the cell-vertex scheme is fewer than that for the cell-centered scheme, and fewer equations need to be solved, thus the convergence rate is faster.
- By Fourier analysis, for the same grid, larger size of control volume is beneficial to convergence rate. Because the size of control volume for a cell-vertex scheme is larger than the size of control volume for a cell-centered scheme, so the convergence rate of a cell-vertex scheme is much faster than that of a cell-centered scheme.
Conclusions
The cell-vertex scheme for unstructured triangular mesh is not widely used in FVM due to the complexity of program implementation. However, there are some advantages in both numerical accuracy and convergence rate using this scheme.
The cell-vertex scheme for unstructured triangular mesh is not widely used in FVM due to the complexity of program implementation. However, there are some advantages in both numerical accuracy and convergence rate using this scheme.
- On the same triangular mesh, the cell-vertex scheme is more accurate than the cell-centered scheme, in other words, fewer cells are required to achieve the same accuracy for a cell-vertex scheme.
- On the same triangular mesh, the convergence of the cell-vertex scheme is much faster than that of the cell-centered scheme, thus much computation time can be saved.
An Unstructured Grids-based Discretization Method in Cylindrical Coordinates Systems
Cylindrical symmetrical problems are usually involved in the calculations of heat transfer and fluid flow. In many cases, due to the symmetry of the computation domain and the solution of the physical problem, many practical problems could be simplified from three-dimensional ones to two-dimensional ones.
Here, an unstructured grids-based discretization method, in the framework of a finite volume approach, is proposed for the solution of the convection–diffusion equations in r–z coordinates, and especially an accurate calculation method of the control volumes is presented.
Cylindrical symmetrical problems are usually involved in the calculations of heat transfer and fluid flow. In many cases, due to the symmetry of the computation domain and the solution of the physical problem, many practical problems could be simplified from three-dimensional ones to two-dimensional ones.
Here, an unstructured grids-based discretization method, in the framework of a finite volume approach, is proposed for the solution of the convection–diffusion equations in r–z coordinates, and especially an accurate calculation method of the control volumes is presented.
1. General finite volume discretization method for unstructured grids
Flexibility of unstructured grids allows accurate representation of complex geometry. These techniques could be used for discrete feature modeling and flow simulation in the near-well region. Due to the flexible features, the discretization of partial differential equations using finite volume method for unstructured grids is complicated and challenging.
In order to reduce the discretization error caused by mesh non-orthogonality, we corrected the flux with the aid of higher-order total variational diminishing (TVD) scheme and adopted momentum interpolation method (MIM) for pressure-velocity coupling.
Figure (a) shows the discretized equation of the steady-state dimensionless convection–diffusion equation and figure (b) illustrates the basic principles of higher-order TVD scheme.
Flexibility of unstructured grids allows accurate representation of complex geometry. These techniques could be used for discrete feature modeling and flow simulation in the near-well region. Due to the flexible features, the discretization of partial differential equations using finite volume method for unstructured grids is complicated and challenging.
In order to reduce the discretization error caused by mesh non-orthogonality, we corrected the flux with the aid of higher-order total variational diminishing (TVD) scheme and adopted momentum interpolation method (MIM) for pressure-velocity coupling.
Figure (a) shows the discretized equation of the steady-state dimensionless convection–diffusion equation and figure (b) illustrates the basic principles of higher-order TVD scheme.
![图片](/uploads/2/5/4/3/25435923/8416062.png?384)
2. An accurate calculation method of unstructured control volumes in a two-dimensional cylindrical coordinate system
As seen from the figure (on the right) of control volumes in a two-dimensional cylindrical system ((a) structured, (b) unstructured), the calculation of unstructured control volume could not be determined by the same procedures as structured grids’ and require complicated procedures.
Through analysis, there are only two situations (situation 1 and situation 2) in total. We developed the combination and split method to determine the control volume of unstructured grid.
Figures (a) and (b) below illustrate the usage of the combination and split method as well as the stereogram of unstructured control volume ((a) situation 1, (b) situation 2).
As seen from the figure (on the right) of control volumes in a two-dimensional cylindrical system ((a) structured, (b) unstructured), the calculation of unstructured control volume could not be determined by the same procedures as structured grids’ and require complicated procedures.
Through analysis, there are only two situations (situation 1 and situation 2) in total. We developed the combination and split method to determine the control volume of unstructured grid.
Figures (a) and (b) below illustrate the usage of the combination and split method as well as the stereogram of unstructured control volume ((a) situation 1, (b) situation 2).
![图片](/uploads/2/5/4/3/25435923/8278112.png?516)
Numerical Example
In this section, the natural convection in an irregular cylindrical cavity is investigated. Due to the symmetry of the physical domain and solutions, only the two-dimensional domain shown in figure (a) on the right needs to be concerned about.
To validate the correctness of the proposed method, the irregular domain is mapped by structured grids and unstructured grids respectively (see figure (b) on the right).
In this section, the natural convection in an irregular cylindrical cavity is investigated. Due to the symmetry of the physical domain and solutions, only the two-dimensional domain shown in figure (a) on the right needs to be concerned about.
To validate the correctness of the proposed method, the irregular domain is mapped by structured grids and unstructured grids respectively (see figure (b) on the right).
Results and Analysis
Figure (a) below gives the comparison of temperature fields calculated by the structured grids-based and the unstructured grids-based discretization method, with two groups of computation parameters. It can be seen that the results of the two methods are almost the same with each other. The unstructured grids present very good flexibility to the irregular domain and thus lead to more accurate results than that of structured grids for an irregular domain.
The results of the proposed method are also compared with those calculated by the famous commercial software, FLUENT. Figure (b) below presents the comparison of the results calculated by the proposed method and FLUENT in one specific case of this example. It is found that the result calculated by the proposed method agrees well with that calculated by FLUENT.
Figure (a) below gives the comparison of temperature fields calculated by the structured grids-based and the unstructured grids-based discretization method, with two groups of computation parameters. It can be seen that the results of the two methods are almost the same with each other. The unstructured grids present very good flexibility to the irregular domain and thus lead to more accurate results than that of structured grids for an irregular domain.
The results of the proposed method are also compared with those calculated by the famous commercial software, FLUENT. Figure (b) below presents the comparison of the results calculated by the proposed method and FLUENT in one specific case of this example. It is found that the result calculated by the proposed method agrees well with that calculated by FLUENT.
Conclusions
We have proposed an unstructured grids-based discretization method for the convection–diffusion equations in cylindrical coordinates, in the framework of a finite volume approach. Numerical results have validated the correctness of the proposed method. Though the proposed discretization method is performed only on unstructured triangular grids, it could be readily extended to that on an unstructured quadrilateral grids system.
The study provides great convenience for the application of unstructured grids in a two-dimensional cylindrical coordinate system, leading to the flexibility of the discretization method for the irregular domains of any shapes.
We have proposed an unstructured grids-based discretization method for the convection–diffusion equations in cylindrical coordinates, in the framework of a finite volume approach. Numerical results have validated the correctness of the proposed method. Though the proposed discretization method is performed only on unstructured triangular grids, it could be readily extended to that on an unstructured quadrilateral grids system.
The study provides great convenience for the application of unstructured grids in a two-dimensional cylindrical coordinate system, leading to the flexibility of the discretization method for the irregular domains of any shapes.