A Study of the Mathematical Aspects of Some Selected Conformal Mapping Airfoil Design Theories and Their Applications

Abstract

This paper discusses the mathematical aspects that undergird two conformal mappings that are commonly used in airfoil design and well-known to most aeronautical engineering practitioners, namely, the Joukowski and Schwarz-Christoffel transformations. It orients the reader with the necessary analytical complex-analysis tools required in the derivation and application of these mappings. The paper explores how the Schwartz Christoffel theorem transforms a unit circle into an airfoil that is envisaged as a polygon divided into n linear segments called panels. Both transformations are well catalogued, with more emphasis being placed on the Joukowski mapping, which is more commonly used.  To accomplish this, the generalisation of the Joukowski transformation is progressively developed from the simple mapping of the circle into an ellipse. Subsequently, the desirable cambering effect, which is attained both by varying the parameter of transformation λ and offsetting the centre of the transformation, is explored and validated for a circle of radius 1.24 centred alternatively at -0.24+0i, and -0.22+0.28i. This transformation is presented diagrammatically to enhance the visualisation of how the Joukowski transformation maps a cylinder (or circle) into an airfoil. In the end, the Joukowski transformation in particular is reaffirmed as an important framework for the achievement of optimal characteristics in the preliminary design phase of wings and airfoils.  Throughout the paper, the fluid flow is assumed to be two-dimensional, irrotational under gravitational influence, incompressible, steady, and inviscid, and to have negligible dissipative viscous forces. All boundary layers and wakes are assumed to be vanishingly thin.