A Modern Look at Conformal Mapping Including Multiply Connected Regions

With the advent of large-scale digital computers, numerical solution of the transonic flow equations in two dimensions has proven feasible. In such calculations it is convenient, and often crucial, to map the region of in- terest conformally onto a simple domain so that a simple finite difference system can be employed. This paper updates a well-known airfoil conformal mapping method (Theodorsen and Garrick) with modern techniques, greatly enhancing the mapping speed and accuracy while simplifying the analysis. Also, a powerful new class of conformal transformations is introduced, and applied to a two-element airfoil. N the calculation of transonic flow over airfoils it has proven convenient to map the infinite region of the flow exterior to an airfoil conformally onto a finite region, such as the interior of a circle, as in Sells,l Melnik and Ives,2 Jameson,3 and Bauer, Garabedian, and Korn.4 Such a mapping places a fine grid in the physical plane where needed, while retaining a uniform grid in the (con- formally mapped) computational plane. The gradients of flow properties in the computational plane are smaller than in the physical plane, so that finite difference approximations in the computational plane are more accurate than in the physical plane. This allows the use of a rather coarse grid. It is easier to apply boundary conditions in the mapped plane than it is in the physical plane, as the boundaries are coordinate lines in the mapped plane. The conformal mapping ensures that the nonlinear partial differential equations in the computational plane are only slightly more complicated than those in the physical plane.1'5 To take advantage of the above con- siderations, there is a need for conformal mapping techniques that are simple and rapid. This paper presents a conformal mapping technique which is at least an order of magnitude faster than conventional mapping techniques. In analogy to the Fourier transform techniques utilized here, the present work can be considered a fast conformal mapping. This paper also introduces a powerful new class of con- formal transformations. These new transformations are an extension of the well-known von Karman-Trefftz trans- formation to multiply-conn ected regions, and can be used in the analysis of flow over multielement airfoils. They have the property that any number N of airfoil elements are simultaneously (with one application of the transformation) mapped to TV near circles. One member of this new class of transformations is used in mapping the region exterior to a two-element airfoil onto the region between two concentric circles. By combining the known solution for incompressible flow between two con- centric circles6 with this two-element mapping, we can very simply solve for the inviscid incompressible flow over a two- element air foil.