Convergence of lift and drag predictions by a Morino panel method (VSAERO)

Abstract

CONVERGENCE results from both two-dimensional and three-dimensional versions of a Morino-type code are presented an analyzed. In both two and three dimensions it is found that reasonably accurate lift and moment predictions are obtained using modest numbers of nominally uniform-sized panels, but induced drag predictions are seriously inaccurate and converge at best very slowly with increasing density. Two ways are found to improve accuracy and convergence of drag predictions: 1) use cosine spacing for chordwise distribution of panels on foil sections, or 2) deduce lift and drag from the far-field wake by a classical Trefftzplane analysis. Of the two methods, the far-field has superior accuracy and convergence properties and, indeed, produces highly accurate and stable drag predictions even from quite coarse panelizations. It is also shown that large benefits in accuracy and/or savings in computational effort are possible by use of Richardson extrapolation applied to results from two or more relatively coarse (but systematically related) panelizations. Contents Although methods have been extensively employed and validated for aerodynamic lift and moment calculations, they have a reputation for being inaccurate or unreliable with respect to drag. The work reported here was stimulated by attempts to apply the code VSAERO to calculate the lift and induced drag of complex wing-body configurations representing sailboat hull, keel, and rudder geometries,1 for which induced drag (effective span) is the characteristic that most strongly impacts performance. (Note that there exist various versions of VSAERO. The three-dimensional code used in this study, a public-domain version obtained directly from NASA in 1984 and currently distributed by COSMIC, may differ from commercial versions.) Conventional VSAERO drag predictions were found to be highly sensitive to details and density of panelization and in poor agreement with experiments. These studies, undertaken to discover the sources and magnitude of the errors, led eventually to usably accurate drag predictions by two methods that are believed to be applicable to most three-dimensional codes. Any code is a discrete approximation to a continuous flow: a mostly smooth body surface is approximated by an assemblage of triangular or quadrilateral panels, and continuous flow quantities (e.g., singularity strengths and pressure) are approximated by simple distributions over each panel. As the subdivision of the body is made successively finer, the discretized model approximates the true body geometry and the continuous flow more closely and so, presumably, computed flow quantities such as lift and drag will converge toward the exact continuous solution. The rate of convergence is of considerable practical interest because the computational effort of a method increases rapidly with the number TV of panels. Using a finer panelization than necessary is, therefore, very expensive in terms of time and computer resources. Also, if reliable rates of convergence can be established for some systematic way of subdividing the panels, a projection to infinite density is possible from two or more relatively coarse panelizations using Richardson extrapolation. VSAERO2'3 is a low-order three-dimensional subsonic code for general configurations. In common with many current production codes, for its potential-flow calculations it adopts the Morino formulation4 with constant known source and unknown doublet panels. In the solution phase, a Morino code sets up and solves a large system of linear equations for the unknown doublet densities on the panels. In the phase, surface velocities, hence pressures, are obtained by local differencing of the doublet densities; these pressures are multiplied by areas and normals and summed to obtain total forces. This near-field force calculation is referred to in the sequel as panel pressure analysis (PPA). A principal tool for investigating the sources of drag error was VSA2D, a two-dimensiona l incompressible code based on the same singularity model, the same integral equation, and the same pressure, force, and moment calculations as VSAERO. VSA2D alternatively evaluates lift and drag from a far-field (FFA), which is extremely simple in two dimensions. Numerical results are given for two airfoils for which exact analytic results are available for comparison: a biconvex section and a Joukowsky section. Some lift and drag results are plotted in Fig.l against I/TV, for both uniform and cosine spacings on the 10% biconvex section, with TV ranging from 6 to 80.