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ABSTRACT
A new decomposition method for the discretization of anisotropic diffusion term is developed. The method is a generalization of the optimum decomposition practice adopted in discretizing the isotropic diffusion flux. The new approach is applied in conjunction with the well-known semi-implicit and recently developed modified implicit nonlinear diffusion schemes and used for discretizing the anisotropic diffusion term. The resulting discretization methods are used for solving several anisotropic diffusion problems to compare the performance of the new decomposition technique with the standard one. Results generated demonstrate the virtues of the new method, which leads to a reduction in the CPU times needed for convergence by percentages reaching a level as high as 70%.
Nomenclature
| C | = | main grid point at an element centroid |
| dCF | = | vector joining the two points C and F |
| dCF | = | magnitude of dCF |
| eCF | = | unit vector in the direction of dCF |
| E | = | distance vector in the direction of dCF |
| = | distance vector in the direction of dCF | |
| E | = | magnitude of E |
| = | magnitude of | |
| F | = | neighbor of element C |
| f | = | face |
| K | = | diffusion coefficient tensor |
| kxx, kxy | = | diffusion coefficients |
| NC | = | location used in the calculation of the nonorthogonal part of the diffusion flux |
| NF | = | location used in the calculation of the nonorthogonal part of the diffusion flux |
| Q | = | source term in conservation equation |
| S | = | surface vector |
| = | modified surface vector | |
| S | = | magnitude of S |
| = | magnitude of | |
| n | = | unit vector in the direction of S |
| = | unit vector in the direction of | |
| t | = | unit vector in the direction of T |
| T | = | vector equal to S − E |
| = | vector equal to | |
| V | = | cell volume |
| Greek symbols | = | |
| ϕ | = | general variable |
| = | averaging factors | |
| = | averaging factors satisfying Eq. (11) | |
| θ | = | rotation angle |
| ξ, η | = | curvilinear coordinates |
| Subscripts | = | |
| C | = | refers to main grid point |
| f | = | refers to element face |
| F | = | refers to the F grid point |
| Superscripts | = | |
| T | = | refers to the transpose of a vector |
| — | = | refers to an interpolated value |
Nomenclature
| C | = | main grid point at an element centroid |
| dCF | = | vector joining the two points C and F |
| dCF | = | magnitude of dCF |
| eCF | = | unit vector in the direction of dCF |
| E | = | distance vector in the direction of dCF |
| = | distance vector in the direction of dCF | |
| E | = | magnitude of E |
| = | magnitude of | |
| F | = | neighbor of element C |
| f | = | face |
| K | = | diffusion coefficient tensor |
| kxx, kxy | = | diffusion coefficients |
| NC | = | location used in the calculation of the nonorthogonal part of the diffusion flux |
| NF | = | location used in the calculation of the nonorthogonal part of the diffusion flux |
| Q | = | source term in conservation equation |
| S | = | surface vector |
| = | modified surface vector | |
| S | = | magnitude of S |
| = | magnitude of | |
| n | = | unit vector in the direction of S |
| = | unit vector in the direction of | |
| t | = | unit vector in the direction of T |
| T | = | vector equal to S − E |
| = | vector equal to | |
| V | = | cell volume |
| Greek symbols | = | |
| ϕ | = | general variable |
| = | averaging factors | |
| = | averaging factors satisfying Eq. (11) | |
| θ | = | rotation angle |
| ξ, η | = | curvilinear coordinates |
| Subscripts | = | |
| C | = | refers to main grid point |
| f | = | refers to element face |
| F | = | refers to the F grid point |
| Superscripts | = | |
| T | = | refers to the transpose of a vector |
| — | = | refers to an interpolated value |