Unsteady 2D potential-flow forces on a thin variable geometry airfoil undergoing arbitrary motion

In this report analytical expressions for the unsteady 2D force distribution on a variable geometry airfoil undergoing arbitrary motion are derived under the assumption of incompressible, irrotational, inviscid flow. The airfoil is represented by itscamberline as in classic thin-airfoil theory, and the deflection of the airfoil is given by superposition of chordwise deflection mode shapes.

It is shown from the expressions for the forces, that the influence from the shed vorticity in the wake isdescribed by the same time-lag for all chordwise 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 ofDuhamel 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 stabilityanalysis. The analytical expressions for the forces simplify to all previously known steady and unsteady thin-airfoil solutions.

Apart from the obvious applications within active load control/reduction, the current theory can be used for variousapplications which up to now have been possible only using much more computational costly methods. The propulsive performance of a soft heaving propulsor, and the influence of airfoil camberline elasticity on the flutter limit are two computationalexamples given in the report that highlight this feature.