Most flying animals produce aerodynamic forces by flapping their wings back and forth with a complex wingbeat pattern. The fluid dynamics that underlies this motion has been divided into separate aerodynamic mechanisms of which rotational lift, that results from fast wing pitch rotations, is particularly important for flight control and manoeuvrability. In our study we focus on the generation of these rotational forces during the wing-reversal and come to a new and remarkably simpel rotational lift model.

Insects fly by flapping there wings back and forth, contrary to birds which often move there wings up and down. To understand this unique wingbeat pattern scientist have tried to link the motion of the wing to the forces the insect generates ^{Ellington1984}. In the end this will enable us to gain insight in how an insect flies, but also how the morphology of the insect influences the force generation. Essentially this helps us understand how an insects stays in the air.

Before we can say anything about the force generation we need to know how an insect moves. Unfortunately a flying insect moves really fast, but luckily for us high-speed camera’s can be used to slow down this motion. Lets look at a great movie on how a fruit-fly flies captured by ^{Muijres2014}

It might be difficult to see but the insect moves the wings to the front of its body (horizontal to the ground) under an angle of around \(45^o\), when it can no longer move forward it rotates its wing around the length of the wing and then move back again.

This rotation is the focus of our manuscript. It has been discovered some time ago by ^{Dickinson1999} that pitching a wing around its longitudinal axis that is moving forward with a certain stroke velocity generates a force. These forces are known as rotational forces. However, as mentioned by ^{Bomphrey2017} it appears that mosquitoes also generates rotational forces at wing-reversal, when the wing is not moving forward or backward, so what gives?

The idea from ^{Nakata2015} is that there is an additional rotational lift force, called “rotational drag”, which is linked to the rotational velocity about the longitudinal axis of the wing. This means that an insect **can** generate rotational forces at wing-reversal.

Based on the this notion from all the previous mentioned authors we set out to do a parametric study, varying the forward velocity, wing shape, and longitudinal rotational velocity systematically to understand how these additional rotational forces are generated.

For this study we used computational fluid mechanics, which is a method that can simulate the movement of the air if the motion of the animal is known. The solver which we used is open-source and can be found here: IBAMR ^{Bhalla2013}.

From our results we noticed that indeed ^{Nakata2015} was correct, that insects can generate additional rotational forces at wing-reversal. We also found that the geometry influences these forces in a very systematic way. The additional rotational forces are influenced by the asymmetry in the shape of the wing along the longitudinal axis. This means that for a symmetrical wing no additional forces are generated!

From our parametric study we also where able to create a very simple rotational lift model

\begin{equation} F_{\mathrm{rotational}} = C_{F, \mathrm{rotational}} \rho \left(\sqrt{S_{xx}S_{yy}} \omega_{\mathrm{stroke}} \omega_{\mathrm{pitch}} + S_{x|x|} \omega_{\mathrm{pitch}}^2 \right) \end{equation}

where the coefficient \(C_{F,\mathrm{rotational}} \approx \frac{2}{3}\pi\), \(\rho\) the density of the air, \(\sqrt{S_{xx}S_{yy}}\) the symmetric second moment of area, \(\omega_{\mathrm{stroke}}\) the stroke velocity, \(\omega_{\mathrm{pitch}}\) the pitch velocity and \(S_{x|x|}\) the asymmetric second moment of area and finally \(F_{\mathrm{rotational}}\) the rotational forces, which are defined perpendicular to the wing surface.

This asymmetric second moment of area \(S_{x|x}\) is essentially a measure of how much wing surface is above or below the pitch axis. This means that if we move the pitch axis more towards the leading edge (introducing the chord-wise offset) the wing-asymmetry increases and so do the rotational forces. This also holds true the other way around, if we decrease the pitch axis, moving it closer to the symmetry axis of the wing the rotational forces decrease.

These rotational forces are of importance for the insect because it is been shown by ^{Muijres2014} that the kinematics is adapted at wing-reversal when a maneuver is performed.

[Ellington1984] Ellington, The aerodynamics of hovering insect flight. IV. aerodynamic mechanisms, *Philosophical transaction royal society London*, (1984). ↩

[Muijres2014] Muijres, Elzinga, Melis & Dickinson, Flies evade looming targets by executing rapid visually directed banked turns, *Science*, **344**, 172-177 (2014). ↩

[Dickinson1999] Dickinson, Lehmann & Sane, Wing roation and the aerodynamic basis of insect flight, *Science*, **284**, 1954-1960 (1999). ↩

[Bomphrey2017] Bomphrey, Nakata, Phillips & Walker, Smart wing rotation and trailing-edge vortices enable high frequency mosquito flight, *Nature*, **544**, 92-96 (2017). ↩

[Nakata2015] Nakata, Liu Hao & Bomphrey, A CFD-informed quasi-steady model of flapping-wing aerodynamics, *Journal of Fluid Mechanics*, **783**, 323-343 (2015). ↩

[Bhalla2013] Bhalla, Bale, Griffith & Patankar, A unified mathematical framework and an adaptive numerical method for fluid–structure interaction with rigid, deforming, and elastic bodies, *Journal of computational physics*, **250**, 446-476 (2013). ↩