Catastrophe-Theory-Based Modeling of Airfoil-Stall Boundary at Low Reynolds Numbers

Abstract

Airfoil stall at low Reynolds numbers is a complex nonlinear dynamic phenomenon, which is characterized by catastrophe and hysteresis. It is difficult but important to mathematically describe the stall points under different Reynolds numbers. In this paper, taking the clockwise hysteresis as an example, a modeling method is proposed to describe the boundary of static airfoil stall according to the topological properties and physical characteristics of airfoil stall. Through numerical simulations, the lift characteristics of an airfoil at different Reynolds numbers are computed, and it is found that Reynolds number can affect not only the catastrophe and hysteresis of airfoil stall, but also the size of the hysteresis loop. Next, the static stall at low Reynolds numbers and the dynamic behavior described by the cusp-catastrophic model are proved to have a similarity in spatial topology and physical properties. According to the topological invariant rules, the topological-mapping relationship between the catastrophe-point set of the cusp-catastrophic model and the stall points of the airfoil stall is established through the development of an accurate topological-transformation function. Consequently, the catastrophe lines described by the cusp-catastrophic model can be used to represent the static-stall boundary of the airfoil at different Reynolds numbers. The effect of model description is verified by comparing the model-predicted values with the simulated values, and the error is found to be less than 1%.