On the adoption of velocity variable and grid system for fluid flow computation in curvilinear coordinates

Abstract The issues of adopting the velocity components as dependent velocity variables, including the Cartesian and culvilinear ones, for the Navier-Stokes flow computations are investigated. The viewpoint advocated is that a numerical algorithm should preferably honor both the physical conservation law in differential form and the geometric conservation law in discrete form. It is demonstrated that with the curvilinear velocity vectors the curvatures of the grid lines introduce extra source terms into the governing equations. With the Cartesian velocity vector, on the other hand, the governing equations in curvilinear coordinates can retain the full conservation-law form and honor the physical conservation laws. The nonlinear combinations of the metric terms also cause the algorithms based on curvilinear velocity components to be more difficult to satisfy the geometric conservation law and, hence, more sensitive to grid skewness effect. For the combined utilization of the Cartesian velocity vector and the staggered grid arrangement, the implications of spurious pressure oscillation arising from the 90° turning are discussed. It is demonstrated that these spurious oscillations can possibly appear only under a very specific circumstance, namely, the meshes in the region with 90° turning must be parallel to the Cartesian coordinates and of uniform spacing along coordinates; otherwise no spurious oscillations can appear. Several flow solutions for domain with 90° and 360° turnings are presented to demonstrate that satisfactory results can be obtained by using the Cartesian velocity components and the staggered grid arrangement.