C h a p t e r 6
airfoil. The lift on the airfoil generally followed the trends predicted by the simple thin airfoil theory models, unless the wakes of the generators interfered substantially.
This limited the usefulness of both generators. In some cases, the pitching airfoil changed the flow around the test article to such an extent that the average forces were significantly different before and after pitching. If the pitching airfoil was near the midline of the tunnel, it also reduced the coherent vortex shedding from the test article when it was at moderate angles of attack. This limits its position to be far in the y-direction from the test article. The heaving gust generator was conceived to reduce the effect of the generator’s wake on the test article, and was shown to not affect the test article when it was kept by the side of the tunnel. When the plate passed the midline, its wake impacted the test article twice, with significant effects on the flow and forces on the test article. Avoiding this interaction limited the polarity of the shed vortices: those with positive circulation below the airfoil, and negative above it.
The angle of attack of the airfoil had a significant impact on the time needed for the forces to transition from their perturbed levels to their final states. When the airfoil was atα= 0°, the forces approached their final state quickly, on a timescale of 5-10 tc, which is similar to that of the Wagner function. When the airfoil was almost statically stalled atα = 10°, large perturbations from the gust could cause the flow to reattach, resulting in recovery timescales of up to 20tc. This is consistent with the timescales reported in control of separated flows [12] and rapidly accelerated plates [47].
The heaving plate yielded one result unavailable with the pitching plate: the timescale of resumption of vortex shedding from the airfoil atα= 5°, after a strong perturbation, was 15tc. The wake of the pitching gust generator had overwhelmed the development of the vortex shedding behavior in its experiments.
When the wakes did not significantly interfere with the test article the theoretical models provided reasonable estimates of the initial lift peak. After this peak, the simulations did not properly capture the evolution of the forces, as the simulations lacked models of the boundary layers and separated flow around the airfoil. The more complex models, the unsteady panel method and the extended Tchieu-Leonard model, did not provide substantially more accurate estimates of the lift than the simple point-vortex model with the Wagner function. The simplest model with a wake provided a useful balance between computational cost and physical relevance.
It is comforting to see that the simple models of unsteady thin airfoil theory provide
accurate models of the interaction between an airfoil and a vortical gust.
6.2 Future Work
The limitations of the two investigated gust generators suggest that a more com- plicated approach to gust generation could be useful. A combination of the two methods may combine the advantages of both, while eliminating most of their dis- advantages. Since the simplest analytical models are based on linear thin airfoil theory, they can be additively combined. The combined motion will then be a mix of both pitching and heaving. At its simplest, consider an impulsive change in heaving velocity,∆S= S2−S1, and angle of attack,∆α= α2−α1, att = 0. In this case, consider S to be a signed velocity, so that positive values indicate motion in the+y direction, and vice-versa. Thus the combination of Equations 2.6 and 2.13 yields
Γv = π
2Uc(∆α−∆S). (6.1)
The previously analyzed heaving plate system had S1 = −S2 = S and ∆α = 0.
The pitching airfoil system had S1 = S2 = 0 and ∆α = α2. The combination of the two methods allows for a gust generator that advances and retracts to spare the test article its permanent influence, while being able to generate vortices of either polarity regardless of being above or below the test article. This simple analysis ignores any additional complications, such as the pitching location or any lead or lag between the motions, which may have an effect on the resulting gust. Further research could investigate optimal approaches to gust generation in this manner.
This research could also be used in conjunction with sensing and control schemes to reduce the effect of a vortical gust on a wing. Since the analytical methods provided a reasonable estimate of the initial response of the forces to a gust, this is a valid framework for advancement. For example, an approaching vortex with negative circulation could be countered by an increase in the wing’s angle of attack, or a downward deflection of a trailing-edge flap. These are simplistic responses, but may merit further investigation.
These simple simulations were not able to fully capture the gust response. Further research may develop simple models that can capture the behavior of the reattaching and separating flows around the airfoils, as well as the vortex’s interaction with the wake. These could yield better predictions of the forces, and thus be more useful
for future control schemes. Such models may require more detailed simulations to examine subtle phenomena in the airfoil’s boundary layer.