Analytical and Computational Investigations of Airfoils Undergoing High-Frequency Sinusoidal Pitch and Plunge Motions at Low Reynolds Numbers

Abstract

Current interests in Micro Air Vehicle (MAV)technologies call for the development of aerodynamic-design tools that will aid in the design of more efficient platforms that will also have adequate stability and control for flight in gusty environments. Influenced largely by nature MAVs tend to be very small, have low flight speeds, and utilize flapping motions for propulsion. For these reasons the focus is, specifically, on high-frequency motions at low Reynolds numbers. Toward the goal of developing design tools, it is of interest to explore the use of elementary flow solutions for simple motions such as pitch and plunge oscillations to predict aerodynamic performance for more complex motions.

In the early part of this research, a validation effort was undertaken. Computations from the current effort were compared with experiments conducted in a parallel, collaborative effort at the Air Force Research Laboratory (AFRL). A set of pure-pitch and pure-plunge sinusoidal oscillations of the SD7003 airfoil were examined. Phase-averaged measurements using particle image velocimetry in a water tunnel were compared with computations using two flow solvers: i) an incompressible Navier-Stokes Immersed Boundary Method and ii) an unsteady compressible Reynolds-Averaged Navier-Stokes (RANS) solver. The motions were at a reduced frequency of $k = 3.93$, and pitch-angle amplitudes were chosen such that a kinematic equivalence in amplitudes of effective angle of attack (from plunge) was obtained. Plunge cases showed good qualitative agreement between computation and experiment, but in the pitch cases, the wake vorticity in the experiment was substantially different from that predicted by both computations. Further, equivalence between the pure-pitch and pure-plunge motions was not attained through matching effective angle of attack.

With the failure of pitch/plunge equivalence using equivalent amplitudes of effective angle of attack, the effort shifted to include pitch-rate and wake-effect terms through the use of analytical methods including quasi-steady thin-airfoil theory (QSTAT) and Theodorsen's theory. These theories were used to develop three analytical approaches for determining pitch motions equivalent to plunge motions. A study of variation in plunge height was then examined and followed by a study of the effect of rotation point using the RANS solver. For the range of plunge heights studied, it was observed that kinematic matching between plunge and pitch using QSTAT gave outstanding similarities in flow field, while the matching performed using Theodorsen's theory gave the best equivalence in lift coefficients for all cases. The variation of rotation point revealed that, for the given plunge height, with rotation point in front of the mid-chord location, all three schemes matched flow-field vorticity well, and with rotation point aft of the mid-chord no scheme matched vorticity fields. However, for all rotation points (except for the mid-chord location), CFD prediction of lift coefficients from the Theodorsen matching scheme matched the lift time histories closely to CFD predictions for pure-pitch.

Combined pitch and plunge motions were then examined using kinematic parameters obtained from the three schemes. The results showed that QSTAT nearly cancels the vortices emanating from the trailing edge. Theodorsen's matching approach was successful in generating a lift that was close to constant over the entire cycle. Additionally this approach showed the presence of the reverse Karman vortex sheet through the wake. Combined pitch/plunge motions were then analyzed, computationally and experimentally, with a non-zero mean angle of attack. All computational results compared excellently with experiments, capturing vorticity production on the airfoil's surface and through the wake. Lift coefficient through a cycle was shown to tend toward a constant using Theodorsen's parameters, with the constant being dependent on the initial angle of attack. This result points to the possibility of designing an unsteady motion to match a given flight-condition requirement.