Investigation into the Unsteady Response of Airfoils due to Small-Scale Motion

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Catlett, Matthew Ryan
Baeder, James
Unsteady pressures, forces, and pitching moments generated by foils experiencing vibratory motion in an incompressible, attached flow configuration are studied within this work. Specifically, two-dimensional, unsteady potential flow calculations and analysis are developed and performed on various numerically thin and Joukowski foils undergoing variable amplitude, small-scale heaving and pitching motions over a range of reduced frequencies between approximately 0.01 to 100. While the calculations and analysis are performed in the context of potential flow, the wide range of reduced frequencies are intended in order to be as widely applicable as possible, for either aerodynamic of hydrodynamic configurations. Given that substantially large reduced frequencies are being considered, a set of criteria are established based on the product of the reduced frequency and freestream Mach number in order to help ensure the applicability of potential flow to either experimental results or higher fidelity numerical results. Calculated results are compared directly to predictions from implementing the Theodorsen model, which treats foils as infinitely thin, flat plates that shed a planar sheet of vorticity. The effects of relaxing these seemingly strict conditions are explored, and the particular terms which control the unsteady responses are identified and discussed. This involves the consideration of Joukowski foils of varying finite thickness and allowing for the wake of shed vorticity to convect according to the specifics of the unsteady velocity field. For sinusoidal disturbance motion of increasing heaving and pitching amplitude and increasing reduced frequencies the shed wake is seen to become quite non-planar and to form coherent vortex structures. Despite this wake behavior, the normalized foil responses at the disturbance reduced frequency are seen to be largely unaffected. However, non-negligible responses are generated across a wide range of other frequencies which are separate from the specific reduced frequency of the disturbance motion. Potential flow calculations for symmetric Joukowski foils of varying thickness show that there is marginal effect of foil thickness on the unsteady foil responses at reduced frequencies less than one for both heaving and pitching disturbance motions. For higher reduced frequency conditions however, the unsteady foil responses are seen to vary relative to the Theodorsen model in specific instances. For high reduced frequency heaving motion of Joukowski foil profiles with varying thickness, the unsteady pitching moment response, and not the unsteady lift, is seen to vary substantially in both its magnitude and phase properties relative to that of the flat foil. A specific augmenting expression is developed for this behavior through analysis within the potential flow framework. In a very similar manner and approach it is found for pitching disturbance motion about the foil mid-chord, that for variable thickness foils the unsteady lift response, and not the pitching moment, also varies in its magnitude and phase properties relative to the flat foil. A specific augmenting expression is also developed for this configuration, which leverages the fact that the unsteady flow field is defined by explicitly known expressions. Lastly, unsteady potential flow calculations and analysis are also presented which specifically concentrate on the unsteady streamwise force response. While the unsteady properties of the response of this force component is often not a focus, it does exhibit interesting properties and is closely connected to the mean streamwise force response, as well as the unsteady lift response. A predictive expression is developed for the magnitude of the streamwise force response. This expression compares very favorably to the unsteady potential flow calculations performed within these efforts, and is also used successfully to assess the published results from several distinct references.