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ABSTRACT
Two new extremely robust, fully implicit coupled solution procedures (FICS-1 and FICS-2) are presented for the ultimate solution of the notorious velocity–pressure coupling problem arising in incompressible fluid flow problems. Based on a previous idea of the author, the algebraic coupled system of equations resulting from the discretization of the momentum and mass conservation equations is taken in its primitive form. A special incomplete decomposition technique is applied to the block matrix of the algebraic system, requiring only two defect vectors in the defect matrix. With the new mechanism applied, the mass and momentum conservations are satisfied simultaneously at all points of the solution region and at each step of the solution process. In this way, the effect of any change in a dependent variable is sensed immediately at all of the points in the solution region. Contrary to the almost outdated segregated-type approaches, the new procedures do not require any explicit equation for pressure, so that the laborious tasks of formulation and solution of any Poisson-type equations are avoided. The procedures are not pressure-based. They are very simple to formulate and implement. The strong coupling preserved and the full implicitness of the algorithm involved helps in treating the nonlinearities most efficiently through a couple of overall block solutions. Tests on the two procedures presented in this work show that up to at least 20 times faster convergence rates can be achieved, compared with any of the segregated-type procedures, which accounts for a 95% reduction in computing time. The procedures may converge even when no relaxation is applied, but they may converge faster if some optimal relaxation is applied. With these properties, the procedures presented seem to provide a breakthrough in the area of computational fluid mechanics.
Nomenclature
| A | = | discretization coefficients |
| = | coefficient matrix | |
| b | = | right-hand-side coefficients |
| B | = | right-hand-side vector |
| D | = | defect matrix |
| E | = | underrelaxation factor |
| h, f, k, n | = | |
| a, x, b, s | = | |
| g, w, c | = | strength vectors |
| d, y, γ, σ | = | defect vectors |
| L, U | = | strength matrices |
| m | = | number of nodes in x direction |
| n | = | number of nodes in y direction |
| ntot | = | total number of nodes |
| p | = | pressure |
| u, v | = | velocity components |
| x, y | = | space coordinates |
| α | = | underrelaxation parameter |
| ϵ | = | pressure head of entering fluid |
| μ | = | diffusivity |
| ρ | = | density |
| Subscripts | = | |
| e, w, n, s | = | cell faces |
| S, W, P, E, N | = | grid nodes |
| i | = | indices for nodes |
| in | = | indicates inflow |
| out | = | indicates outflow |
| Superscripts | = | |
| exact | = | denotes exact value |
| C | = | denotes continuity |
| p | = | denotes pressure |
| u, v | = | denotes velocity components |
| * | = | denotes preliminary values |
Nomenclature
| A | = | discretization coefficients |
| = | coefficient matrix | |
| b | = | right-hand-side coefficients |
| B | = | right-hand-side vector |
| D | = | defect matrix |
| E | = | underrelaxation factor |
| h, f, k, n | = | |
| a, x, b, s | = | |
| g, w, c | = | strength vectors |
| d, y, γ, σ | = | defect vectors |
| L, U | = | strength matrices |
| m | = | number of nodes in x direction |
| n | = | number of nodes in y direction |
| ntot | = | total number of nodes |
| p | = | pressure |
| u, v | = | velocity components |
| x, y | = | space coordinates |
| α | = | underrelaxation parameter |
| ϵ | = | pressure head of entering fluid |
| μ | = | diffusivity |
| ρ | = | density |
| Subscripts | = | |
| e, w, n, s | = | cell faces |
| S, W, P, E, N | = | grid nodes |
| i | = | indices for nodes |
| in | = | indicates inflow |
| out | = | indicates outflow |
| Superscripts | = | |
| exact | = | denotes exact value |
| C | = | denotes continuity |
| p | = | denotes pressure |
| u, v | = | denotes velocity components |
| * | = | denotes preliminary values |
Notes
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