The theory for an oscillating thin airfoil as derived from the Oseen equations

Abstract

The classical potential solution for the flow about a thin airfoil in either steady or oscillatory motion requires application of the condition, postulated by Kutta, that the fluid velocity be finite at the trailing edge of the airfoil. The Kutta condition derives from the argument that viscous stresses will not allow a flow to turn about a sharp edge. Analytic verification of the validity of this condition, of particular interest in the unsteady case, has not previously been obtained. The problem is treated here by utilizing the Oseen formulation for viscous flow. The solution thus obtained approaches small-perturbation potential flow at a large distance from the airfoil and retains a qualitatively correct representation of the rotational flow near the airfoil. By simply assuming that the resultant force on the airfoil is finite, it is shown that the Kutta condition must apply in the limit of vanishing viscosity.The first-order corrections, for large Reynolds number, to the lift and moment on an oscillating airfoil are explicitly determined. The effect of the Oseen approximation on the applicability of the numerical results remains to be established.