On the quasi-steady aerodynamics of normal hovering flight part I: the induced power factor

Abstract

An analytical treatment to quantify the losses captured in the induced power factor, k, is provided for flapping wings in normal hover, including the effects of non-uniform downwash, tip losses and finite flapping amplitude. The method is based on a novel combination of actuator disc and lifting line blade theories that also takes into account the effect of advance ratio. The model has been evaluated against experimental results from the literature and qualitative agreement obtained for the effect of advance ratio on the lift coefficient of revolving wings. Comparison with quantitative experimental data for the circulation as a function of span for a fruitfly wing shows that the model is able to correctly predict the circulation shape of variation, including both the magnitude of the peak circulation and the rate of decay in circulation towards zero. An evaluation of the contributions to induced power factor in normal hover for eight insects is provided. It is also shown how Reynolds number can be accounted for in the induced power factor, and good agreement is obtained between predicted span efficiency as a function of Reynolds number and numerical results from the literature. Lastly, it is shown that for a flapping wing in hover k owing to the non-uniform downwash effect can be reduced to 1.02 using an arcsech chord distribution. For morphologically realistic wing shapes based on beta distributions, it is shown that a value of 1.07 can be achieved for a radius of first moment of wing area at 40% of wing length.

Optimum hovering wing planform

“…This is shown in Fig. 2 through plotting the downwash and circulation distributions for this chord distribution using the lifting line blade model in [21]. The downwash and circulation distributions produced are very near to the constant and elliptic distributions respectively.…”

“…(9) This shows that the elliptic chord distribution leads to a constant induced angle of attack distribution for both parallel and revolving translations, which is, to the authors at least, a delightful result. The constant sectional velocity distribution associated with parallel translation motion (fixed wing) leads in turn to a constant downwash velocity distribution and thus a unity induced power factor, where the induced power factor (denoted as k ind ) is the ratio of the actual induced power to minimum ideal induced power for a given lift [21][22][23][24]. On the other hand, the linear downwash distribution of the elliptic chord for a revolving wing leads to a k ind value of 1.13 (i.e.…”

“…On the other hand, the linear downwash distribution of the elliptic chord for a revolving wing leads to a k ind value of 1.13 (i.e. 13% more induced power compared to the ideal uniform downwash induced power to produce that lift [21,22]). …”

“…Downwash and circulation distributions are normalised using the maximum value of each distribution. Downwash and circulation distributions are evaluated based on the lifting line blade theory, for details see reference [21].…”

“…Wing twist is zero in each case. The value of k ind is evaluated based on the method developed in reference [21].…”