THREE-DIM ENSIONAL DIAGONAL CARTESIAN METHOD FOR INCOMPRESSIBLE FLOWS INVOLVING COMPLEX BOUNDARIES

A diagonal Cartesian method for the three-dimensional simulation of incompressible fluid flows over complex boundaries is presented in this article. The method is derived utilizing the superposition of the finite analytic solutions of a linearized two-dimensional convection-diffusion equation in Cartesian coordinates. The complex boundary is approximated with a structured grid in a series of calculation planes which are perpendicular to the x, y, and z coordinate axis. In the calculation plane, both Cartesian grid lines and diagonal line segments are used. It is observed that this geometry approximation is more accurate than the traditional sawtooth method. Mass conservation on complex boundaries is enforced with an appropriate pressure boundary condition. The method, which utilizes cell-vertex nodes on a staggered grid, uses boundary velocity information to avoid the specification of pressure values on boundaries. An enlarged control-volume method is introduced for the mass conservation and the pressure boundary condition on complex boundaries. The conservation of momentum on complex boundaries is enforced through the use of three-dimensional 19, 15, 11, or 7-point finite analytic elements. The proposed diagonal Cartesian method is verified by the solution of a rotated lid-driven cavity flow. It is shown that this diagonal Cartesian method predicts the fluid flow very well.