Unsteady Thin-Airfoil Theory Revisited: An Approximate Analytical Solution and the Self-Similar Wagner Effect in Viscous Flows

Abstract

An approximate analytical solution for the unsteady lift of a thin airfoil with a general unsteady motion is derived from a viscous-flow perspective, where the wake vortex-sheet strength is given in an explicit convolution-type expression as an approximate solution of the Wagner integral equation. For validation, this analytical solution is applied to the Wagner and Theodorsen problems, giving the explicit integral forms of the Wagner and Theodorsen functions as the reduced cases. Further, this analytical solution is applied to the starting flow with a finite timescale in the generalized Wagner problem, revealing the self-similarity of the re-normalized circulatory lift coefficient and its equivalence to the re-normalized Wagner function in a finite time domain. More importantly, the self-similar Wagner effect is found in numerical simulation of the flow over a starting flat-plate airfoil at low Reynolds numbers even when the flow is moderately separated. This self-similarity represents the Reynolds-number-invariance.