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ABSTRACT
IDEAL proposed by the present author is an efficient segregated algorithm for solving the incompressible fluid flow and heat transfer problems. However, its convergence rate is greatly influenced by the under-relaxation factor. The convergence rate under an optimum under-relaxation factor is dozens of times quicker than that under the most unfavorable under-relaxation factor. To lessen the influence, the IDEAL algorithm combined with a fuzz control method, called IDEAL+FC, is introduced to automatically regulate the values of the under-relaxation factor for accelerating the iteration convergence. Finally, it is demonstrated that IDEAL+FC is superior to IDEAL in terms of convergence rate and robustness. The rapid convergence rate can be achieved by IDEAL+FC even if the initial under-relaxation factor is the most unfavorable value.
Nomenclature
| a | = | coefficient in the discretized equation |
| b | = | constant term in the discretized equation |
| Cp | = | specific heat, J · kg−1 · K−1 |
| d | = | maximum residual |
| D | = | tube outer diameter, m |
| e | = | ratio of the residuals between two successive iteration levels |
| f | = | friction factor |
| p | = | pressure, Pa |
| R | = | fuzzy relation |
| Re | = | Reynolds number |
| S′u,S′v,S′w S′T | = | source term introduced by the grid nonorthogonality |
| T | = | temperature, K |
| u, v, w | = | velocity components in x, y, z directions, m · s−1 |
| U, V, W | = | contravariant velocity components in ξ, η, ζ directions, m · s−1 |
| y | = | twist ratio |
| α | = | under-relaxation factor |
| ϕ | = | general variable |
| η | = | dynamic viscosity, Pa · s |
| λ | = | thermal conductivity, W · m−1 · K−1 |
| μ | = | membership function |
| ν | = | kinematic viscosity, m2 · s−1 |
| θ | = | inclination angle, ° |
| ρ | = | density, kg · m−3 |
| ξ, η, ζ | = | nonorthogonal curvilinear coordinates |
| Δα | = | change of the under-relaxation factor |
| Subscript | = | |
| e | = | referring to the ratio of the residuals |
| e, w, n, s, t, b | = | grid interface |
| n | = | current iteration level |
| n-1 | = | previous iteration level |
| nb | = | neighboring grid points |
| P | = | main grid point |
| T | = | referring to temperature |
| u, v, w | = | referring to u, v, w velocities |
| Δα | = | referring to the change of the under-relaxation factor |
| Superscript | = | |
| NB, NS | = | negative big, negative small |
| PB, PM, PS | = | positive big, positive medium, positive small |
| T | = | referring to temperature |
| u, v, w | = | referring to u, v, w velocities |
| 0 | = | previous iteration |
Nomenclature
| a | = | coefficient in the discretized equation |
| b | = | constant term in the discretized equation |
| Cp | = | specific heat, J · kg−1 · K−1 |
| d | = | maximum residual |
| D | = | tube outer diameter, m |
| e | = | ratio of the residuals between two successive iteration levels |
| f | = | friction factor |
| p | = | pressure, Pa |
| R | = | fuzzy relation |
| Re | = | Reynolds number |
| S′u,S′v,S′w S′T | = | source term introduced by the grid nonorthogonality |
| T | = | temperature, K |
| u, v, w | = | velocity components in x, y, z directions, m · s−1 |
| U, V, W | = | contravariant velocity components in ξ, η, ζ directions, m · s−1 |
| y | = | twist ratio |
| α | = | under-relaxation factor |
| ϕ | = | general variable |
| η | = | dynamic viscosity, Pa · s |
| λ | = | thermal conductivity, W · m−1 · K−1 |
| μ | = | membership function |
| ν | = | kinematic viscosity, m2 · s−1 |
| θ | = | inclination angle, ° |
| ρ | = | density, kg · m−3 |
| ξ, η, ζ | = | nonorthogonal curvilinear coordinates |
| Δα | = | change of the under-relaxation factor |
| Subscript | = | |
| e | = | referring to the ratio of the residuals |
| e, w, n, s, t, b | = | grid interface |
| n | = | current iteration level |
| n-1 | = | previous iteration level |
| nb | = | neighboring grid points |
| P | = | main grid point |
| T | = | referring to temperature |
| u, v, w | = | referring to u, v, w velocities |
| Δα | = | referring to the change of the under-relaxation factor |
| Superscript | = | |
| NB, NS | = | negative big, negative small |
| PB, PM, PS | = | positive big, positive medium, positive small |
| T | = | referring to temperature |
| u, v, w | = | referring to u, v, w velocities |
| 0 | = | previous iteration |
Acknowledgments
This work was supported by the National Natural Science Foundation of China (51476054) and the Program for New Century Excellent Talents in University (NCET-13-0792) and the BIPT-POPME-2015.
