The two-dimensional subsonic, piecewise continuous-kernel function method used for studying either oscillatory or steady flows is extended in the present work to three-dimensi onal problems involving finite-span wings. The work treats questions associated with the choice of spanwise pressure polynomials, spanwise collocation points, and numerical integration techniques that need be faced by this method. A subsequent paper contains results which confirm the accuracy of the method, its rapid convergence, and its very high efficiency in terms of computational time. The method is tested for a limited class of geometrical discontinuities (i.e., at the wing root only). A third paper contains additional results which relate to a wider class of problems associated with geometrical discontinuities. EOMETRICAL discontinuities have become common in modern airplane wings, including not only control surface deflections but also wing chord discontinuities such as those existing at the root of a delta wing (discontinuity in the first derivative of the chord along the span), at leading-edge extensions, or at wing surface break points. Since these geometrical discontinuities lead to pressure singularities at the same geometrical locations, it is necessary to know the exact form of these pressure singularities if the use of the kernel function method (KFM) is contemplated. The lattice methods (i.e., the vortex or doublet) can successfully cope with unknown pressure singularities if their location is known, but they require a relatively large number of unknowns (boxes) for convergence which at times leads to a relatively large residual error at the converged values.1'2 In Refs. 1 and 2, a different method is proposed which represents the pressure distribution by a set of piecewise continuous polynomials spanning the regions between adjoining singularities (also referred to as boxes) and employs the KFM for solution of the pressure coefficients. It is shown in Refs. 1 and 2 where this two-dimensional problem was treated that such an ap- proach, referred to as the piecewise continuous-kernel func- tion method (PCKFM), has the ability to treat pressure discontinuities in a manner similar to the doublet-lattice method, with the added accuracy and rapid convergence characteristics of the kernel function method. In addition, it is not essential to determine the nature or form of the pressure singularities that might exist along some of the boundaries forming each box. However, to accelerate convergence, pressure singularities are assumed to be known only along the boundaries of the wing; or more specifically, the form of the leading-edge (LE), trailing-edge (TE), and wing-tip pressure singularities are assumed to be known and are treated in the analysis. All other pressure singularities are ignored during the analysis and their consideration is limited to the deter- mination of the boundaries for the different boxes. The basic problems associated with the two-dimensional PCKFM were treated in Refs. 1 and 2. These problems in- cluded the determination of orthogonal pressure polynomials for boxes with different known pressure singularities along their boundaries, the determination of the collocation points associated with the assumed pressure polynomials, the determination of the desired number of boxes, and the number of orthogonal polynomials required in each box. These problems will be addressed again while attempting to extend the PCKFM to wings with finite spans. Additional problems arising in three-dimensi onal flow configurations involve the formulation of numerical techniques which are required for the successful application of the method. These techniques, which are useful beyond the methods described in this work, will be developed herein. Wing-Tip Pressure Singularity and Associated
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Nov 1, 1992
Sheshagiri K Hebbar + 1
Open Access