Diagonal Cartesian method for modeling of incompressible flows over complex boundaries

A "diagonal Cartesian method" is proposed for the simulation of incompressible fluid flow over complex boundaries. A structured grid is utilized for the sake of simplicity. The method approximates the complex boundaries using both Cartesian grid lines and diagonal line segments. A corresponding automatic and problem-independent grid generation method is formulated for complex geometries. A number of examples show that the geometrical approximation is significantly more accurate than the traditional saw-tooth method even though the order of accuracy is not formally increased. A pressure boundary condition is also developed to enforce mass conservation near the complex boundaries. The method, based on cell-centered nodes on a non-staggered grid, uses boundary velocity information to avoid specification of pressure values on the boundaries. The proposed treatment for the pressure boundary condition introduces an "enlarged control volume method". Conservation of momentum at the complex boundaries is enforced through the Finite Analytic method, using 9-point and 5-point elements. Non-zero velocity boundary conditions at moving boundaries are also analyzed. The proposed diagonal Cartesian method is verified by solving a lid-driven cavity flow. It is shown that the method predicts the fluid flow very well. The handling of the pressure boundary condition was further tested on laminar flow past a backward-facing step including both Neumann and Dirichlet velocity boundary conditions. The method is also applied to two channel-like more complex flows: a grooved channel flow such as might be encountered in applications of Micro Electro Mechanical Systems, and the flow within a casting mould. The results suggest that the diagonal Cartesian method might be useful for such practical applications.