Miller Laura A, Peskin Charles S
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA.
J Exp Biol. 2005 Jan;208(Pt 2):195-212. doi: 10.1242/jeb.01376.
In this paper, we have used the immersed boundary method to solve the two-dimensional Navier-Stokes equations for two immersed wings performing an idealized 'clap and fling' stroke and a 'fling' half-stroke. We calculated lift coefficients as functions of time per wing for a range of Reynolds numbers (Re) between 8 and 128. We also calculated the instantaneous streamlines around each wing throughout the stroke cycle and related the changes in lift to the relative strength and position of the leading and trailing edge vortices. Our results show that lift generation per wing during the 'clap and fling' of two wings when compared to the average lift produced by one wing with the same motion falls into two distinct patterns. For Re=64 and higher, lift is initially enhanced during the rotation of two wings when lift coefficients are compared to the case of one wing. Lift coefficients after fling and during the translational part of the stroke oscillate as the leading and trailing edge vortices are alternately shed. In addition, the lift coefficients are not substantially greater in the two-winged case than in the one-winged case. This differs from three-dimensional insect flight where the leading edge vortices remain attached to the wing throughout each half-stroke. For Re=32 and lower, lift coefficients per wing are also enhanced during wing rotation when compared to the case of one wing rotating with the same motion. Remarkably, lift coefficients following two-winged fling during the translational phase are also enhanced when compared to the one-winged case. Indeed, they begin about 70% higher than the one-winged case during pure translation. When averaged over the entire translational part of the stroke, lift coefficients per wing are 35% higher for the two-winged case during a 4.5 chord translation following fling. In addition, lift enhancement increases with decreasing Re. This result suggests that the Weis-Fogh mechanism of lift generation has greater benefit to insects flying at lower Re. Drag coefficients produced during fling are also substantially higher for the two-winged case than the one-winged case, particularly at lower Re.
在本文中，我们使用浸入边界法求解了二维纳维-斯托克斯方程，用于模拟两个浸入式机翼执行理想化的“拍打与抛掷”冲程以及“抛掷”半冲程的情况。我们计算了一系列雷诺数（Re）在8至128之间时每个机翼的升力系数随时间的变化。我们还计算了整个冲程周期内每个机翼周围的瞬时流线，并将升力的变化与前缘和后缘涡旋的相对强度和位置联系起来。我们的结果表明，与一个机翼以相同运动产生的平均升力相比，两个机翼在“拍打与抛掷”过程中每个机翼的升力产生可分为两种不同模式。对于Re = 64及更高值，当将升力系数与一个机翼的情况进行比较时，在两个机翼旋转期间升力最初会增强。抛掷后以及冲程平移部分期间的升力系数会随着前缘和后缘涡旋交替脱落而振荡。此外，双翼情况下的升力系数并不比单翼情况下大很多。这与三维昆虫飞行不同，在三维昆虫飞行中，前缘涡旋在每个半冲程中都始终附着在机翼上。对于Re = 32及更低值，与一个机翼以相同运动旋转的情况相比，每个机翼在机翼旋转期间的升力系数也会增强。值得注意的是，与单翼情况相比，双翼抛掷后平移阶段的升力系数也会增强。实际上，在纯平移期间，它们比单翼情况高出约70%。在抛掷后的4.5弦长平移过程中，对冲程整个平移部分进行平均时，双翼情况下每个机翼的升力系数比单翼情况高35%。此外，升力增强随着Re的减小而增加。这一结果表明，韦斯-福格升力产生机制对在较低Re下飞行的昆虫更有益。抛掷过程中产生的阻力系数在双翼情况下也比单翼情况下高得多，尤其是在较低Re时。
