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ABSTRACT
This article presents a new element-based finite-volume discretization approach for the solution of incompressible flow problems on co-located grids. The proposed method, called the method of proper closure equations (MPCE), employs a proper set of physically relevant equations to constrain the velocity and pressure at integration points. These equations provide a proper coupling between the nodal values of pressure and velocity. The final algebraic equations are not segregated in this study and are solved in a fully coupled manner. To show the applicability and performance of the method, it is tested on several steady two-dimensional laminar-flow benchmark cases. The results indicate that the method simulates the fluid flow in complex geometries and on nonorthogonal computational grids accurately. Also, it is shown that the method is robust in the sense that it does not require severe underrelaxation even at relatively high Reynolds numbers. In each test case, the required underrelaxation parameter, the number of iterations, and the corresponding CPU time are reported.
Nomenclature
| a | = | wavy wall amplitude |
| = | influence coefficients for example in Eq. (38) | |
| A | = | area of control surface, matrix of coefficients |
| bU, bV, bP | = | vectors of known values |
| Cf | = | Skin-friction coefficient |
| = | influence coefficients, for example in Eq. (58) | |
| F | = | flow term |
| Gm | = | influence coefficient in Eq. (44) |
| h | = | nondimensional uniform grid spacing |
| H | = | height |
| = | influence coefficients, for example in Eq. (47) | |
| = | influence coefficients, for example in Eq. (51) | |
| L | = | length, width |
| N | = | bi-linear shape functions and total number of nodes |
| p | = | pressure |
| P | = | dimensionless pressure |
| r | = | radius |
| Re | = | Reynolds number |
| Res | = | residual |
| s, t | = | local coordinates |
| u, v | = | velocity components |
| U, V | = | dimensionless velocity components |
| = | velocity vector | |
| x, y | = | Cartesian coordinates |
| X, Y | = | dimensionless Cartesian coordinates |
| ∇ | = | gradient |
| ⟦ ⟧ | = | maximum value |
| θ | = | inclination angle |
| μ | = | dynamic viscosity |
| ρ | = | density |
| ϕ | = | a general scalar variable |
| ω | = | relaxation parameter |
| Subscripts | = | |
| 0 | = | reference value |
| avg | = | average |
| d | = | down |
| e | = | end point of the wavy wall |
| i | = | inner |
| i, j, m, n, p, q | = | dummy indices |
| ip | = | integration point |
| l | = | left |
| lid | = | lid-driven |
| max | = | maximum |
| o | = | outer |
| r | = | right |
| rms | = | root mean square |
| u | = | up |
| s | = | start point of the wavy wall |
| w | = | wall |
| Superscripts | = | |
| con | = | continuity |
| exa | = | exact |
| mom | = | momentum |
| n | = | iteration level |
| num | = | numerical |
| P | = | pressure |
| U, V | = | velocity components |
| U-conv | = | X-momentum convection |
| U-conv-U,-V, … | = | influence coefficients in X-momentum convection |
| U-diff | = | X-momentum diffusion |
| U-diff-U,-V, … | = | influence coefficients in X-momentum diffusion |
| u-, U-mom | = | x and X momentum |
| U-press | = | X-momentum pressure |
| U-press-U,-V, … | = | influence coefficients in X-momentum pressure |
| v-, V-mom | = | y and Y momentum |
| u-, U-mom | = | x and X momentum |
Nomenclature
| a | = | wavy wall amplitude |
| = | influence coefficients for example in Eq. (38) | |
| A | = | area of control surface, matrix of coefficients |
| bU, bV, bP | = | vectors of known values |
| Cf | = | Skin-friction coefficient |
| = | influence coefficients, for example in Eq. (58) | |
| F | = | flow term |
| Gm | = | influence coefficient in Eq. (44) |
| h | = | nondimensional uniform grid spacing |
| H | = | height |
| = | influence coefficients, for example in Eq. (47) | |
| = | influence coefficients, for example in Eq. (51) | |
| L | = | length, width |
| N | = | bi-linear shape functions and total number of nodes |
| p | = | pressure |
| P | = | dimensionless pressure |
| r | = | radius |
| Re | = | Reynolds number |
| Res | = | residual |
| s, t | = | local coordinates |
| u, v | = | velocity components |
| U, V | = | dimensionless velocity components |
| = | velocity vector | |
| x, y | = | Cartesian coordinates |
| X, Y | = | dimensionless Cartesian coordinates |
| ∇ | = | gradient |
| ⟦ ⟧ | = | maximum value |
| θ | = | inclination angle |
| μ | = | dynamic viscosity |
| ρ | = | density |
| ϕ | = | a general scalar variable |
| ω | = | relaxation parameter |
| Subscripts | = | |
| 0 | = | reference value |
| avg | = | average |
| d | = | down |
| e | = | end point of the wavy wall |
| i | = | inner |
| i, j, m, n, p, q | = | dummy indices |
| ip | = | integration point |
| l | = | left |
| lid | = | lid-driven |
| max | = | maximum |
| o | = | outer |
| r | = | right |
| rms | = | root mean square |
| u | = | up |
| s | = | start point of the wavy wall |
| w | = | wall |
| Superscripts | = | |
| con | = | continuity |
| exa | = | exact |
| mom | = | momentum |
| n | = | iteration level |
| num | = | numerical |
| P | = | pressure |
| U, V | = | velocity components |
| U-conv | = | X-momentum convection |
| U-conv-U,-V, … | = | influence coefficients in X-momentum convection |
| U-diff | = | X-momentum diffusion |
| U-diff-U,-V, … | = | influence coefficients in X-momentum diffusion |
| u-, U-mom | = | x and X momentum |
| U-press | = | X-momentum pressure |
| U-press-U,-V, … | = | influence coefficients in X-momentum pressure |
| v-, V-mom | = | y and Y momentum |
| u-, U-mom | = | x and X momentum |