In Pursuit of Grid Convergence for Two-Dimensional Euler Solutions

摘要

Grid-convergence trends of two-dimensional Euler solutions are investigated. The airfoil geometry under study isnbased on the NACA0012 equation. However, unlike the NACA0012 airfoil, which has a blunt base at the trailingnedge, the study geometry is extended in chord so that its trailing edge is sharp.The flowsolutions use extremely- high-nquality grids that are developedwith the aid of theKarman–Trefftz conformal transformation. The topology of eachngrid is that of a standard O-mesh. The grids naturally extend to a far-field boundary approximately 150 chordnlengths away fromthe airfoil.Each quadrilateral cell of the resultingmesh has an aspect ratio of one.The intersectingnlines of the grid are essentially orthogonal at each vertex within the mesh. A family of grids is recursively derivednstartingwith the finestmesh.Here, each successively coarser grid in the sequence is constructed by eliminating everynother node of the current grid, in both computational directions. In all, a total of eight grids comprise the family,withnthe coarsest-to-finestmeshes having dimensions of 32 u0001 32–4096 u0001 4096 cells, respectively.Note that the finest grid innthis family is composed of over 16million cells, and is suitable for 13 levels of multigrid. The geometry and grids arenall numerically defined such that they are exactly symmetrical about the horizontal axis to ensure that a nonliftingnsolution is possible at zero degrees angle-of-attack attitude.Characteristics of threewell-known flowsolvers (FLO82,nOVERFLOW, and CFL3D) are studied using a matrix of four flow conditions: (subcritical and transonic) byn(nonlifting and lifting). The matrix allows the ability to investigate grid-convergence trends of 1) drag with andnwithout lifting effects, 2) drag with and without shocks, and 3) lift and moment at constant angles-of-attack. Resultsnpresented herein use 64-bit computations and are converged to machine-level-zero residuals. All three of the flownsolvers have difficultymeeting this requirement on the finestmeshes, especially at the transonic flowconditions. Somenunexpected results are also discussed. Take for example the subcritical cases. FLO82 solutions do not reachnasymptotic grid convergence of second-order accuracy until the mesh approaches one quarter of a million cells.nOVERFLOW exhibits at best a first-order accuracy for a central-difference stencil. CFL3D shows second-ordernaccuracy for drag, but only first-order trends for lift and pitching moment. For the transonic cases, the order ofnaccuracy deteriorates for all of the methods. A comparison of the limiting values of the aerodynamic coefficients isnprovided. Drag for the subcritical cases nearly approach zero for all of the computational fluid dynamics methodsnreviewed. These and other results are discussed.