Navier-Stokes calculations using Cartesian grids. I - Laminar flows

A finite-volume formulation for the Navier-Stokes equations using Cartesian grids is used to study flows past airfoils. In addition to the solution of the complete equations, solutions for two simplified versions of the governing equations were obtained and compared with those using body-fitted grids. Results are presented for two airfoil sections, NACA 0012 and RAE 2822, for a range of Mach numbers, angles of attack, and Reynolds numbers. It is shown that the results are highly dependent on the smoothness of the surface grid. Without such smoothness, the skin friction and pressure converge to nonuniform distributions. On the other hand, when surface cells with smoothly varying areas are used, the results compare favorably with calculations employing body-fitted grids. URRENTLY, much of the work in computational fluid dynamics is focused on the solution of the Navier-Stokes equations for increasingly complex airfoil and wing configura- tions. Some of the issues being examined in these solutions are the efficiency of the algorithms, the accuracy of the solutions, and the generation of grids for complex geometries. Cartesian grids are being pursued as a simple alternative to complicated grid generation techniques for general configurations. The successful implementation of Cartesian grids for the solution of the Euler equations1 demonstrated the simplicity and utility of these grids, especially when multi-element configurations were considered. More recently, use of Cartesian grids was extended to the solution of the Euler equations for flowfields around three-dimensional geometries.2 The goal of the present work is to show the feasibility of using Cartesian grids in the solution of the Navier-Stokes equations for complex two-dimensional configurations. Be- cause of the many issues that were uncovered by the investi- gation, it was decided to limit the scope of the paper to laminar flows. The algorithm employed is similar to that used in Refs. 1 and 2 and is based on the explicit Runge-Kutta time- stepping scheme of Jameson et al. 3 Modifications pertaining to numerical damping were examined. Moreover, the crucial influence of surface cells on the pressure and skin friction is investigated in detail. A cell-centered scheme is employed. For such a scheme it is necessary to estimate the pressure and the various components of the stress tensor and the heat flux vector at the wall. The wall pressure was estimated by extrapolation.1'2 Two different methods for determining wall stresses are used, and their effect on the solution is discussed. In addition to the solution of the full equations, two sim- plified versions of the governing equations are considered.