A three-dimensional unstructured grid arbitrarily shaped element method is developed for solving the Navier-Stokes equations. The proposed method is generally applicable to arbitrarily shaped elements and, thus, offers the potential to unify many of the different grid topologies into a single formulation Examples of grid topologies include a structured hexahedral mesh; unstructured tetrahedral mesh; hybrid mesh using a combination of hexahedra, tetrahedra, prisms, and pyramids; Cartesian mesh with elements (cubes) cut by now boundaries; and mixed arbitrary multiblock meshes. The concept of such an arbitrary element method is developed and implemented into a pressure-based finite volume solver. It utilizes a collocated and cell-centered storage scheme. Expressions for second-order discretizatious of the convection and diffusion terms are derived and presented for an arbitrarily shaped element. The developed code, as a first step, is applied to two selected three dimensional viscous flows using a structured hexahedral mesh and an unstructured tetrahedral mesh. It is demonstrated that the concept of the arbitrarily shaped element method can be viable and efficient.
