Deng Jian, Caulfield C P
Department of Mechanics, Zhejiang University, Hangzhou 310027, People's Republic of China.
BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, United Kingdom and Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Apr;91(4):043017. doi: 10.1103/PhysRevE.91.043017. Epub 2015 Apr 24.
We report the inherently three-dimensional linear instabilities of a propulsive wake, produced by a flapping foil, mimicking the caudal fin of a fish or the wing of a flying animal. For the base flow, three sequential wake patterns appear as we increase the flapping amplitude: Bénard-von Kármán (BvK) vortex streets; reverse BvK vortex streets; and deflected wakes. Imposing a three-dimensional spanwise periodic perturbation, we find that the resulting Floquet multiplier |μ| indicates an unstable "short wavelength" mode at wave number β=30, or wavelength λ=0.21 (nondimensionalized by the chord length) at sufficiently high flow Reynolds number Re=Uc/ν≃600, where U is the upstream flow velocity, c is the chord length, and ν is the kinematic viscosity of the fluid. Another, "long wavelength" mode at β=6 (λ=1.05) becomes critical at somewhat higher Reynolds number, although we do not expect that this mode would be observed physically because its growth rate is always less than the short wavelength mode, at least for the parameters we have considered. The long wavelength mode has certain similarities with the so-called mode A in the drag wake of a fixed bluff body, while the short wavelength mode appears to have a period of the order of twice that of the base flow, in that its structure seems to repeat approximately only every second cycle of the base flow. Whether it is appropriate to classify this mode as a truly subharmonic mode or as a quasiperiodic mode is still an open question however, worthy of a detailed parametric study with various flapping amplitudes and frequencies.
我们报告了由拍打箔片产生的推进尾流的固有三维线性不稳定性，该箔片模仿鱼的尾鳍或飞行动物的翅膀。对于基流，随着拍打幅度的增加，会出现三种连续的尾流模式：贝纳德 - 冯·卡门（BvK）涡街；反向BvK涡街；以及偏转尾流。施加三维展向周期性扰动后，我们发现所得的弗洛凯乘数|μ|表明，在足够高的流动雷诺数Re = Uc / ν≃600时，波数β = 30（或波长λ = 0.21，以弦长无量纲化）处存在不稳定的“短波长”模式，其中U是上游流速，c是弦长，ν是流体的运动粘度。另一种在β = 6（λ = 1.05）处的“长波长”模式在稍高的雷诺数下变得临界，不过我们预计这种模式在物理上不会被观测到，因为至少对于我们所考虑的参数，其增长率总是小于短波长模式。长波长模式与固定钝体阻力尾流中所谓的模式A有一定相似性，而短波长模式的周期似乎约为基流周期的两倍，因为其结构似乎大约仅在基流的每第二个周期重复。然而，将这种模式归类为真正的亚谐波模式还是准周期模式是否合适仍是一个悬而未决的问题，值得针对各种拍打幅度和频率进行详细的参数研究。
