Unsteady loading on an airfoil of arbitrary thickness

Abstract

The unsteady loading on an airfoil of arbitrary thickness is evaluated by using the generalized form of Blasius theorem and a conformal mapping that maps the airfoil surface onto a circle. For a blade vortex interaction the results show that the time history of the unsteady loading is determined by the passage of the vortex relative to the leading edge singularity in the circle plane. The singularity lies inside the circle and moves to a smaller radius as the thickness is increased, causing the unsteady loading pulse to be smoothed. The effect of angle of attack is to move the stagnation point relative to the leading edge singularity and this significantly increases the unsteady lift if the vortex passes on the suction side of the airfoil. These characteristics are different for a step upwash gust, which is considered as a simplified model of a large scale turbulent gust. It is shown that the time history of the magnitude of the unsteady loading is almost completely unaltered by angle of attack for the step gust, but it's direction of action rotates forward by an angle equal to the angle of attack, extending an earlier result by Howe for a flat plate in a turbulent flow to airfoils of arbitrary thickness. However spectral analysis of the gust shows that the high frequency blade response is reduced as the thickness of the airfoil is increased.