Numerical simulation of fluid flows over complex geometrie still remains a challenging problem because (a) it is difficult to accurately approximate the complex boundaries, esp. for three dimensional flows (b) Automation of grid generation is hard to achieve, esp for three dimensional problems with multi-body systems. This research tries to approximate the complex geometries in Cartesian coordinates. Besides Cartesian grid lines the diagonal segments are utilized. Therefore, the complexity of boundary-fitted coordinate or unstructured grid method involving in the flows over complex boundaries can be avoided.

A code is created to approximate the complex geometries and classify the grid node types in Cartesian coordinates. When the geometric object is specified with a set of discrete points, the approximate representation of the object contour is drawn based on the local monotonic principle such that the essential topographical chracter of the object is preserved. The accuracy of the approximated contour on the Cartesian coordinates is estimated by the convergence of contour length E1 and normal distance E2 between the approximated and original contour.

The proposed techniques is illustrated by the approximation of cavity, grooved channel, automobile contour, lake bank and the boundary of porous media. Saw-tooth approximations means the approximations only by Cartesian grid lines while the diagonal approximation is made by the proposed method.

The approximationis of three dimensional complex geometries are shown in above figures. Approximations of two dimensional complex geometries are illustrated in following figures.

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