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ABSTRACT
In this paper, a fully implicit method for the discretization of the diffusion term is presented in the context of the cell-centered finite volume method. The newly developed fully implicit method is denoted by the modified implicit nonlinear diffusion (MIND) scheme. The method is used to solve several isotropic and anisotropic diffusion problems in two-dimensional domains covered with structured (quadrilateral elements) and unstructured (triangular elements) grid systems. The comparison of generated results with similar ones obtained using the semi-implicit scheme demonstrates the superior robustness and accuracy of the MIND scheme and its good convergence characteristics for all types of meshes.
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 |
| = | distance vectors in the direction of dCF | |
| = | magnitude of E, E′, and | |
| F | = | neighbor of element C |
| f | = | face |
| J | = | diffusion flux |
| K | = | diffusion coefficient tensor |
| kxx, kxy | = | diffusion coefficients |
| NC, NF | = | location used in the calculation of the nonorthogonal part of JC and JF, respectively |
| Q | = | source term in conservation equation |
| = | surface and modified surface vectors | |
| = | magnitude of | |
| = | unit vector in the direction of S′ and T, respectively | |
| = | vectors equal to S − E, | |
| V | = | cell volume |
| Greek symbols | = | |
| ϕ | = | general variable |
| δC, δF | = | averaging factors |
| = | averaging factors satisfying Eq. (13) | |
| θ | = | rotation angle |
| Subscripts | = | |
| b | = | refers to boundary |
| C | = | refers to main grid point |
| f | = | refers to element face |
| F | = | refers to the F grid point |
| ∞ | = | refers to free stream value |
| 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 |
| = | distance vectors in the direction of dCF | |
| = | magnitude of E, E′, and | |
| F | = | neighbor of element C |
| f | = | face |
| J | = | diffusion flux |
| K | = | diffusion coefficient tensor |
| kxx, kxy | = | diffusion coefficients |
| NC, NF | = | location used in the calculation of the nonorthogonal part of JC and JF, respectively |
| Q | = | source term in conservation equation |
| = | surface and modified surface vectors | |
| = | magnitude of | |
| = | unit vector in the direction of S′ and T, respectively | |
| = | vectors equal to S − E, | |
| V | = | cell volume |
| Greek symbols | = | |
| ϕ | = | general variable |
| δC, δF | = | averaging factors |
| = | averaging factors satisfying Eq. (13) | |
| θ | = | rotation angle |
| Subscripts | = | |
| b | = | refers to boundary |
| C | = | refers to main grid point |
| f | = | refers to element face |
| F | = | refers to the F grid point |
| ∞ | = | refers to free stream value |
| Superscripts | = | |
| T | = | refers to the transpose of a vector |
| — | = | refers to an interpolated value |
Acknowledgment
The support provided by the American University of Beirut and HSLU in the name of Prof. E. Casartelli is gratefully acknowledged.