Wu Jiang Hao, Zhang Yan Lai, Sun Mao
School of Transportation Science and Engineering, Institute of Fluid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing, China.
J Exp Biol. 2009 Oct;212(Pt 20):3313-29. doi: 10.1242/jeb.030494.
When an insect hovers, the centre of mass of its body oscillates around a point in the air and its body angle oscillates around a mean value, because of the periodically varying aerodynamic and inertial forces of the flapping wings. In the present paper, hover flight including body oscillations is simulated by coupling the equations of motion with the Navier-Stokes equations. The equations are solved numerically; periodical solutions representing the hover flight are obtained by the shooting method. Two model insects are considered, a dronefly and a hawkmoth; the former has relatively high wingbeat frequency (n) and small wing mass to body mass ratio, whilst the latter has relatively low wingbeat frequency and large wing mass to body mass ratio. The main results are as follows. (i) The body mainly has a horizontal oscillation; oscillation in the vertical direction is about 1/6 of that in the horizontal direction and oscillation in pitch angle is relatively small. (ii) For the hawkmoth, the peak-to-peak values of the horizontal velocity, displacement and pitch angle are 0.11 U (U is the mean velocity at the radius of gyration of the wing), 0.22 c=4 mm (c is the mean chord length) and 4 deg., respectively. For the dronefly, the corresponding values are 0.02 U, 0.05 c=0.15 mm and 0.3 deg., much smaller than those of the hawkmoth. (iii) The horizontal motion of the body decreases the relative velocity of the wings by a small amount. As a result, a larger angle of attack of the wing, and hence a larger drag to lift ratio or larger aerodynamic power, is required for hovering, compared with the case of neglecting body oscillations. For the hawkmoth, the angle of attack is about 3.5 deg. larger and the specific power about 9% larger than that in the case of neglecting the body oscillations; for the dronefly, the corresponding values are 0.7 deg. and 2%. (iv) The horizontal oscillation of the body consists of two parts; one (due to wing aerodynamic force) is proportional to 1/cn2 and the other (due to wing inertial force) is proportional to wing mass to body mass ratio. For many insects, the values of 1/cn2 and wing mass to body mass ratio are much smaller than those of the hawkmoth, and the effects of body oscillation would be rather small; thus it is reasonable to neglect the body oscillations in studying their aerodynamics.
当昆虫悬停时，由于扑翼产生的周期性变化的气动力和惯性力，其身体的质心会围绕空气中的某一点振荡，并且其身体角度会围绕一个平均值振荡。在本文中，通过将运动方程与纳维 - 斯托克斯方程耦合来模拟包括身体振荡的悬停飞行。这些方程通过数值求解；通过打靶法获得代表悬停飞行的周期解。考虑了两种模型昆虫，一种是食蚜蝇，另一种是天蛾；前者具有相对较高的振翅频率（n）和较小的翅质量与身体质量比，而后者具有相对较低的振翅频率和较大的翅质量与身体质量比。主要结果如下。（i）身体主要进行水平振荡；垂直方向的振荡约为水平方向振荡的1/6，并且俯仰角的振荡相对较小。（ii）对于天蛾，水平速度、位移和俯仰角的峰峰值分别为0.11U（U是翅膀回转半径处的平均速度）、0.22c = 4毫米（c是平均弦长）和4度。对于食蚜蝇，相应的值分别为0.02U、0.05c = 0.15毫米和0.3度，远小于天蛾的值。（iii）身体的水平运动会使翅膀的相对速度略微降低。结果，与忽略身体振荡的情况相比，悬停时需要更大的攻角，从而需要更大的阻力与升力比或更大的气动功率。对于天蛾，攻角比忽略身体振荡的情况大约大3.5度，比功率大约大9％；对于食蚜蝇，相应的值分别为0.7度和2％。（iv）身体的水平振荡由两部分组成；一部分（由于翅膀气动力）与1/cn²成正比，另一部分（由于翅膀惯性力）与翅质量与身体质量比成正比。对于许多昆虫，1/cn²和翅质量与身体质量比的值远小于天蛾，身体振荡的影响会相当小；因此在研究它们的空气动力学时忽略身体振荡是合理的。
