Aerodynamics Theory of Undulated Wing Sections

An undulated wing section refers to an airfoil whose surface exhibits some form of waviness or undulation. This undulation may result from intentional design to modify the flow characteristics over the airfoil surface, enabling better control and optimizing velocity and pressure distributions. This work presents an exact solution to the problem of 2-D potential flow over undulated wing sections using conformal mapping. A potential flow solution for the flow over a circular cylinder with an undulated surface is first derived. The Joukowsky transformation is then employed to transform this solution to obtain the flow over a family of undulated airfoils. Unlike Theodorsen's theory for arbitrary wing sections, where the problem is reduced to a compact integral equation, the newly derived theory is forward-looking (from undulated cylinders to undulated airfoils) and easy to implement. Expressions for the airfoil surface's velocities and pressure coefficient, lift (circulation) force are derived. The results have been found to depend on both the magnitude of the undulation and its wavenumber. This theory for undulated airfoils is particularly relevant and useful in bioinspired wing applications, especially when seeking to replicate or enhance aerodynamic performance observed in nature.