In Pursuit of Grid Convergence for Two-Dimensional Euler Solutions

Grid-convergence trends of two-dimensional Euler solutions are investigated. The airfoil geometry under study is based on the NACA0012 equation. However, unlike the NACA0012 airfoil, which has a blunt base at the trailing edge, the study geometry is extended in chord so that its trailing edge is sharp. The flow solutions use extremely-high-quality grids that are developed with the aid of the Karman-Trefftz conformal transformation. The topology of each grid is that of a standard O-mesh. The grids naturally extend to a far-field boundary approximately 150 chord lengths away from the airfoil. Each quadrilateral cell of the resulting mesh has an aspect ratio of one. The intersecting lines of the grid are essentially orthogonal at each vertex within the mesh. A family of grids is recursively derived starting with the finest mesh. Here, each successively coarser grid in the sequence is constructed by eliminating every other node of the current grid, in both computational directions. In all, a total of eight grids comprise the family, with the coarsest-to-finest meshes having dimensions of 32 x 32-4096 x 4096 cells, respectively. Note that the finest grid in this family is composed of over 16 million cells, and is suitable for 13 levels of multigrid. The geometry and grids are all numerically defined such that they are exactly symmetrical about the horizontal axis to ensure that a nonlifting solution is possible at zero degrees angle-of-attack attitude. Characteristics of three well-known flow solvers (FLO82, OVERFLOW, and CFL3D) are studied using a matrix of four flow conditions: (subcritical and transonic) by (nonlifting and lifting). The matrix allows the ability to investigate grid-convergence trends of 1) drag with and without lifting effects, 2) drag with and without shocks, and 3) lift and moment at constant angles-of-attack. Results presented herein use 64-bit computations and are converged to machine-level-zero residuals. All three of the flow solvers have difficulty meeting this requirement on the finest meshes, especially at the transonic flow conditions. Some unexpected results are also discussed. Take for example the subcritical cases. FLO82 solutions do not reach asymptotic grid convergence of second-order accuracy until the mesh approaches one quarter of a million cells. OVERFLOW exhibits at best a first-order accuracy for a central-difference stencil. CFL3D shows second-order accuracy for drag, but only first-order trends for lift and pitching moment. For the transonic cases, the order of accuracy deteriorates for all of the methods. A comparison of the limiting values of the aerodynamic coefficients is provided. Drag for the subcritical cases nearly approach zero for all of the computational fluid dynamics methods reviewed. These and other results are discussed.