A modified Leishman-Beddoes model for airfoil sections undergoing dynamic stall at low Reynolds numbers

A modified Leishman-Beddoes[1] model is proposed to predict the aerodynamic load responses of airfoils undergoing dynamic stall at low Reynolds, low Mach numbers and low to very high equivalent reduced pitch rates. The modifications include using Wagner function aerodynamics for the attached flow model, defining and using a time-varying reduced pitch rate, equating the time delays applied to the normal force, the separation point location and the vortex shedding and using Sheng and Galbraith's dynamic stall onset criteria [2]. The steady and unsteady force and pitching moment measurements of three airfoils with distinct stall mechanisms (a flat plate, a NACA0012, and a NACA0018) are used to demonstrate the validity of this new model. Dynamic tests consist of oscillating the airfoils in pitch around the quarter chord with a mean angle A0 = 10 deg and with different prescribed reduced frequencies from k = 0.015, to k = 0.16, and amplitudes from A = 5 deg to 20 deg, at Reynolds numbers of the order of Re=1.8x104. The resulting equivalent reduced pitch rates range from r' = 0.001 to 0.06. The computations of the values of the different model parameters are demonstrated and it is observed that the values of the time delay parameters change continuously and smoothly with time instead of jumping discontinuously at discrete time instances as proposed by Leishman and Beddoes [1]. It is shown that the modified Leishman-Beddoes model is in good agreement with experiments for low to moderate equivalent reduced pitch rates (r'<=0.04). It also significantly improves the dynamic load prediction in this range of pitch rates, in comparison to the original Leishman-Beddoes model. Nevertheless, as for the original model, the modified model presented here becomes inaccurate at the highest equivalent reduced pitch rates of r' > 0.05.