Instead, two approaches, thin-airfoil theory and computational boundary-element (or panel) methods, that can be extended to three-dimensional flows are described. For this reason, it is no longer widely used in aerodynamic design and is not discussed further here. However, the technique cannot be extended to three-dimensional or high-speed flows.
If aerodynamic design involved only two-dimensional flows at low speeds, a design method based on conformal transformation would be a good choice. Advanced versions of the method exploited such modern computing techniques as fast Fourier transforms. Airfoil theory based on conformal transformation became a practical tool for aerodynamic design in 1931, when the American engineer Theodorsen developed a method for airfoils of arbitrary shape, which continued to be developed well into the second half of the twentieth century. Kármán and Trefftz devised a more general conformal transformation, in 1918, that gave a family of airfoils with trailing edges of finite angle. Moreover, all members of this family have a cusped trailing edge, whereas airfoils in practical aerodynamics have trailing edges with finite angles. It applies only to a particular family of airfoil shapes. The magnitude of the circulation is chosen to satisfy the Kutta condition in the z plane.įrom a practical point of view, Joukowski's theory suffers an important drawback. This makes it possible to use the results for the cylinder with circulation (see Section 5.3.10) to calculate the flow around an airfoil. (where C is a parameter), then maps the complex potential flow around the circle in the ζ plane to the corresponding flow around the airfoil in the z plane.
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