Investigations of Lift-Based Pitch-Plunge Equivalence for Airfoils at Low Reynolds Numbers

The limits of linear superposition in two-dimensional high-rate low-Reynolds-number aerodynamics are examined by comparing the lift-coefficient history and flowfield evolution for airfoils undergoing harmonic motions in pure pitch, pure plunge, and pitch―plunge combinations. Using quasi-steady airfoil theory and Theodorsen's formula as predictive tools, pitching motions are sought that produce lift histories identical to those of prescribed plunging motions. It follows that a suitable phasing of pitch and plunge in a combined motion should identically produce zero lift, canceling either the circulatory contribution (with quasi-steady theory) or the combination of circulatory and noncirculatory contributions (with Theodorsen's formula). Lift history is measured experimentally in a water tunnel using a force balance and is compared with two-dimensional Reynolds-averaged Navier―Stokes computations and Theodorsen's theory; computed vorticity contours are compared with dye injection in the water tunnel. Theodorsen's method evinces considerable, and perhaps surprising, resilience in finding pitch-to-plunge equivalence of lift-coefficient―time history, despite its present application to cases in which its mathematical assumptions are demonstrably violated. A combination of pitch and plunge motions can be found such that net lift coefficient is nearly identically zero for arbitrarily high reduced frequency, provided that amplitude is small. Conversely, cancellation is possible at large motion amplitude, provided that reduced frequency is moderate. The product of Strouhal number and nondimensional amplitude is therefore suggested as the upper bound for when superposition and linear predictions remain valid in massively unsteady two-dimensional problems.