本研究首先驗證了在靜止狀態下，平板相較於其他翼型於低雷諾數下不同攻角皆有較好的飛行性能，此結果與Liu結果相符。接著我們進行平板動態擺動的流場模擬，先從上下對稱的運動軌跡著手，軌跡、角度的變化及相位角皆參考Lentink et al.的實驗，相位角的變化有0度、90度、180度、270度，觀察相位角的改變會產生的影響，並觀察升力與阻力係數以及尾流結構的過渡區，發現在相位角90度時，無因次化波長13以下會產生尾流後方大於自由流速的噴流，在相位角180度時，無因次化波長9以下會在平板後方產生推力，因此可得到結論在相位角90度及180度時會因拍動頻率增加而增加推力，在相位角0度時可以發現隨著拍動頻率的增加升力係數變化不大，而阻力係數則會隨著拍動頻率增加而增加，最後在相位角270度時，升力係數隨著拍動頻率的增加力係數也增加。最後在同樣的雷諾數、拍動振福及頻率下改變其軌跡，運動軌跡將參考Zhou et al.，模擬生物拍動鰭或是翅膀時的運動模式，為上下不對稱的運動模式，結果發現在模擬鳥類的軌跡上升力係數提升，並且根據Schouveiler et al.所得到的結論，在拍動全程加上一固定角度，發現在全程加上10度的角度可以得到最高的升力係數；但在龜類拍鰭的軌跡並沒有得到更好的推力，因此改變上衝及下衝時的角度以得到更好的推力，發現在下衝25度及上衝15度時可以得到最高的推力係數。本研究將會以升力係數、阻力係數及尾流結構與相圖的關係來觀察實驗的變化。

In this study, we firstly verified that under static condition, the flat plate has better flight performance than other airfoils at different angles of attack at low Reynolds number, which is in line with the results of Liu. Then we carry out the flow field simulation of the dynamic oscillation of the flat plate. We starts with the symmetric motion trajectory. The trajectory, the change of angle and the phase angle are referred to the experiment of Lentink et al.. The value of phase angle are 0 degree, 90 degree, 180 degree and 270 degree, and we observe the effect of phase angle, the coefficient of lift and drag, and the transition zone of the wake structure. It is found that at phase angle of 90 degrees, a jet larger than the free-flow velocity behind the wake is generated when the non-dimensional wavelength is under 13. At phase angle of 180 degrees, thrust is generated behind the flat plate when the non-dimensional wavelength is under 9. Therefore, it can be concluded that the thrust is increased due to an increase in the flap frequency at phase angles of 90 degrees and 180 degrees. At phase angle of 0 degree, it can be found that the lift coefficient does not change much with increase of the flap frequency, while the drag coefficient increases with increase of flap frequency. Finally, at phase angle of 270 degrees, the lift coefficient increases as the flap frequency increases. The trajectory is changed at the same Reynolds number, flap amplitude and frequency. The motion trajectory will be modeled after Zhou et al., which simulates the motion pattern of a bird or fish flapping its fins or wings, which is an asymmetric motion pattern. The coefficient of lift was found increasing in the simulated bird trajectory. According to the conclusion of Schouveiler et al., the highest coefficient of lift was found to be obtained by adding a fixed angle to the whole flapping process, and it was found that adding an angle of 10 degrees to the whole process can get the highest lift coeffient in this study. However, the trajectory of the fin of the tortoise did not give better thrust, so the angle during the upstroke and downstroke was change for a better thrust, and it was found that the highest thrust coefficient was obtained at 25 degrees downstroke and 15 degrees upstroke. In this study, the experimental changes will be observed in terms of lift coefficient, drag coefficient, and the relationship between the wake structure and the phase diagram.

1. Liu H. 2021. Aerodynamic characteristics of flat plate airfoil at low reynolds numbers. Johns Hopkins University.
2. Lentink D, Muijres FT, Donker-Duyvis FJ, van Leeuwen JL. 2008. Vortex-wake interactions of a flapping foil that models animal swimming and flight. Journal of Experimental Biology. 211(2):267-273.
3. Zhou K, Liu JK, Chen WS. 2018. Numerical and experimental studies of hydrodynamics of flapping foils. J Hydrodyn. 30(2):258-266.
4. Schouveiler L, Hover FS, Triantafyllou MS. 2005. Performance of flapping foil propulsion. Journal of Fluids and Structures. 20(7):949-959.
5. Lentink D, Gerritsma M. 2003. Influence of airfoil shape on performance in insect flight. 33rd AIAA Fluid Dynamics Conference and Exhibit.
6. Williamson CH, Roshko A. 1988. Vortex formation in the wake of an oscillating cylinder. Journal of Fluids and Structures. 2(4):355-381.
7. Berg AM, Biewener AA. 2010. Wing and body kinematics of takeoff and landing flight in the pigeon (columba livia). Journal of Experimental Biology. 213(10):1651-1658.
8. Davenport J, Munks SA, Oxford P. 1984. A comparison of the swimming of marine and freshwater turtles. Proceedings of the Royal society of London Series B Biological Sciences. 220(1221):447-475.
9. Godoy-Diana R, Aider JL, Wesfreid JE. 2008. Transitions in the wake of a flapping foil. Physical Review E. 77(1):5.
10. Menon K, Mittal R. 2019. Flow physics and dynamics of flow-induced pitch oscillations of an airfoil. Journal of Fluid Mechanics. 877:582-613.
11. Van Buren T, Floryan D, Smits AJ. 2019. Scaling and performance of simultaneously heaving and pitching foils. AIAA J. 57(9):3666-3677.
12. Schnipper T, Andersen A, Bohr T. 2009. Vortex wakes of a flapping foil. Journal of Fluid Mechanics. 633:411-423.
13. Andersen A, Bohr T, Schnipper T, Walther JH. 2017. Wake structure and thrust generation of a flapping foil in two-dimensional flow. Journal of Fluid Mechanics. 812:12.
14. Moreira D, Mathias N, Morais T. 2020. Dual flapping foil system for propulsion and harnessing wave energy: A 2d parametric study for unaligned foil configurations. Ocean Eng. 215:11.
15. Ashraf MA, Young J, Lai JCS. 2011. Reynolds number, thickness and camber effects on flapping airfoil propulsion. Journal of Fluids and Structures. 27(2):145-160.
16. Isoda Y, Tanaka Y, Murata S. 2020. Experimental study on the influence of frequency ratio on thrust and vortex structure of a pitching wing in a periodic flow. Advanced Experimental Mechanics. 5:38-43.
17. Mackowski AW, Williamson CHK. 2015. Direct measurement of thrust and efficiency of an airfoil undergoing pure pitching. Journal of Fluid Mechanics. 765:524-543.
18. Bohl DG, Koochesfahani MM. 2009. Mtv measurements of the vortical field in the wake of an airfoil oscillating at high reduced frequency. Journal of Fluid Mechanics. 620:63-88.
19. Garrick IE. 1937. Propulsion of a flapping and oscillating airfoil. NACA Tech. Rep. 567.
20. Young J, Lai JCS. 2004. Oscillation frequency and amplitude effects on the wake of a plunging airfoil. AIAA J. 42(10):2042-2052.
21. Triantafyllou GS, Triantafyllou MS, Grosenbaugh MA. 1993. Optimal thrust development in oscillating foils with application to fish propulsion. Journal of Fluids and Structures. 7(2):205-224.