Novel Approach to Aerodynamic Analysis Using Analytical/Numerical Matching

Analytical/numerical matching (ANM) is a hybrid scheme combining a low-resolution global numerical solution with a high-resolution local solution to form a composite solution. ANM is applied to lifting surfaces in steady po- tential flow to calculate the aerodynamic loading and associated circulation distribution. The solution methodology utilizes overlapping smoothed doublets and local corrections to calculate the doublet strength distribution along the airfoil chord. The global low-resolution solution is calculated numerically using smoothed doublet solutions to the linear potential equation, and converges quickly. Simultaneous local corrections are done with high-resolution local analytical solutions. The global numerical solution is asymptotically matched to the local analytical solutions via a matching solution. The matching solution cancels the global solution in the near field, and cancels the local solution in the far field. The method is very robust, offering insensitivity to control-point location. No explicit wake geometry is assumed; therefore, a fixed- or free-wake model can be used. ANM provides high-resolution calcu- lations from low resolution numerics with analytical corrections, while avoiding the subtlety involving singular integral equations, and their numerical implementation. The approach provides a novel alternative treatment to lifting-surface aerodynamics. ANEL methods,1 or singularity methods, are a cornerstone of modern aerodynamic analysis. These methods construct aero- dynamic flowfields from the superposition of simpler building-block solutions of the governing equations distributed over the body sur- faces. The appropriate strengths of these elementary solutions are determined by the application of surface boundary conditions. Panel methods are used to analyze compressible, steady and unsteady, potential flow over wings and bodies for a variety of applications including performance, loads analysis, aeroelasticity, vibration, in- teractional and interference effects, and acoustics. Although these methods have been in use for a number of years, and have been the subject of much research and refinement, they can possess significant shortcomings. Zero-thickness lifting-surf ace methods can be sensitive to the location of the control points at which the boundary conditions are applied. Certain rules, such as the well- known ~| chord rule for vortex lattice, must be strictly adhered to or incorrect answers will result.2 Panels must be of regular shape and arrangement, partly because of this sensitivity. Problems may arise when panel sizes or orientations vary rapidly, when panels also must be compatible with a separate grid for structural analysis, and when other surfaces or wakes are in close proximity. The analysis of free wakes by these methods poses a particular problem because wake motion leads to irregular distortion of the constituent panels, and their proximity and orientation relative to other panels on aero- dynamics surfaces is difficult to anticipate. Some of these problems have been addressed with higher-order panel techniques3'4 only in the context of steady aerodynamics, however. Finally, there are other difficulties and subtleties associated with the numerical implementation of zero-thickness lifting-surface met- hods. These methods are the numerical embodiment of a singular integral equation having a kernel function whose discretization is open to interpretation, leading to possible ambiguity in the meaning of computed results. For an extensive review of panel methods, the reader may wish to consult the review papers of Erickson,1 Johnson,5 Margason,6 and Landahl and Stark.7 Given the many years of research focused on singularity meth- ods, it should be stressed that the present research involves a novel and innovative reformulation of the fundamental approach to sin- gularity methods, with the goal of alleviating some of the short- comings and difficulties associated with them. The basis for this re- formulation is a new analysis technique called analytical/numerical matching (ANM). The method is applied to lifting surfaces in both two- and three-dimensional steady potential flow to calculate the aerodynamic loading. ANM ANM is a hybrid technique that combines analytical and nu- merical solutions by a matching procedure. ANM allows a global low-resolution numerical solution and a local high-resolution ana- lytical solution to be combined formally by asymptotic matching to construct an accurate composite solution. Both the numerical and the analytical solutions are simpler and more easily obtained than the solution of the original problem, and the overall solution proce- dure is accurate and computationally efficient. The ANM approach provides a high degree of spatial resolution in local areas without great computational burden. ANM is a general analysis method originally developed by D. B. Bliss for application to problems in vortex dynamics and rotor- craft free-wake analysis.8"11 Recently, ANM has been applied to problems involving acoustic radiation and structural-acoustic scat- tering from fluid-loaded structures with discontinuities. In all of these cases, accurate solutions were obtained with significant re- ductions in computational cost. The present work involves further development of ANM and its application to wing aerodynamics. In ANM, an artificial smoothing of the physical problem is intro- duced. The smoothing length scale must be smaller than the large length scales in the problem, but larger than the scale of numeri- cal discretization. Because the smoothing length scale is larger than the scale associated with the numerical discretization, the numerical solution of the smoothed problem is very accurate. However, the ac- tual problem has a physical length scale smaller than the numerical discretization. The local region associated with the small scale is solved separately (usually analytically, but sometimes numerically) as a high-resolution local problem that captures the small scales and rapid variations. This local problem, because of its idealizations, is valid only in the local region. The numerical problem and the local problem are combined by asymptotic matching to form a composite solution. ANM utilizes a matching procedure similar to the method of matched asymptotic