Unsteady two‐dimensional potential‐flow model for thin variable geometry airfoils

Abstract

AbstractIn the present work, analytical expressions for distributed and integral unsteady two‐dimensional forces on a variable geometry airfoil undergoing arbitrary motion are derived under the assumption of incompressible, irrotational, inviscid flow. The airfoil is represented by its camber line as in classic thin‐airfoil theory, and the deflection of the airfoil is given by superposition of chord‐wise deflection mode shapes. It is shown from the expressions for the forces that the influence from the shed vorticity in the wake is described by the same time lag for all chord‐wise positions on the airfoil. This time‐lag term can be approximated using an indicial function approach, making the practical calculation of the aerodynamic response numerically very efficient by use of Duhamel superposition. Furthermore, the indicial function expressions for the time‐lag terms are formulated in their equivalent state‐space form, allowing for use of the present theory in problems employing the eigenvalue approach, such as stability analysis.The analytical expressions for the integral forces can be reduced to Munk's steady and Theodorsen's unsteady results for thin airfoils, and numerical evaluation shows excellent agreement with other unsteady two‐dimensional thin‐airfoil results.Apart from the obvious applications within active load control/reduction, the present theory can be used for various applications which up to the formulation of the present theory have been possible only using much more computational costly methods. An example that highlights this feature is the propulsive performance of a soft heaving propulsor presented in this paper. Copyright © 2010 John Wiley & Sons, Ltd.