本研究旨在使用沉浸邊界法模擬昆蟲飛行動作並探討其飛行之表現。近年來，微型飛行機械 (MAVs) 由於其潛在的軍事用途以及科學應用而備受關注，此外，因為昆蟲的飛行動作具有高效率與機動性強等優點，能做為微型飛行機械設計之參考。然而，大多數先前的研究都是使用單翼模型，忽略了翅膀間的交互作用的影響。因此我們選擇能在低雷諾數下飛行且具有優異飛行表現的果蠅翅膀為模型進行數值模擬，欲模擬的飛行方式有兩種: 真實的果蠅飛行運動 (FF motion) 以及其簡化後的旋轉對稱運動 (SYR motion)。結果顯示翼間的交互作用能對在簡化運動下翅膀互相遠離的過程中產生的升力有接近5 %的提升，並且發現在真實運動中結合翅膀的上升與沉降的動作後對整體飛行表現有更好的提升。

This study aims to investigate the aerodynamic performance of the insect-like flapping flight with direct numerical simulations by using the immersed boundary method. This simulation is used to calculate the lift generation of a flapping wing in a short period of time. In recent years, micro air vehicles (MAVs) have attracted much attention because of their potential military use, also potential applications in science. Moreover, insect-like flapping flight provides a power-efficient and highly maneuverable basis for MAVs. However, most of the previous researches are conducted with the single wing model, instead of considering the effects of wing-wing interaction. Therefore, we simulate wings of fruit flies in particular wing motion because they have an excellent flying performance at low Reynolds number (Re) with computational methods. The results show that the wing-wing interaction and the downward descending motion can enhance the lift due to the stronger leading-edge vortex caused by the lower pressure region between the wings.

Abstract i
List of Figures iv
List of Tables v
1 Introduction 1
1.1 Introduction--------------------------------------1
1.2 Literature Survey---------------------------------2
1.2.1 Immersed Boundary Method------------------------2
1.2.2 PETSc Program-----------------------------------4
1.2.3 Micro Air Vehicle-------------------------------4
1.2.4 Bio-inspired Motion-----------------------------5
1.2.5 Wing-Wing interaction---------------------------7
1.3 Motivation and Objective--------------------------8
2 Numerical Methods-----------------------------------9
2.1 Insect-like Wing Kinematic Motions----------------9
2.1.1 Symmetric Rotating Motion-----------------------9
2.1.2 Realistic Fruit Fly Motion---------------------10
2.2 Immersed-Boundary Method-------------------------10
2.2.1 Mathematical Formulation-----------------------10
2.2.2 Numerical Scheme-------------------------------11
2.2.3 Forcing Strategies-----------------------------12
2.2.4 Determination of Forcing Points----------------14
2.3 Determinations of Lift and Drag Forces-----------14
2.4 Complete Solution Procedure----------------------16
3 Numerical Results----------------------------------24
3.1 Validation---------------------------------------24
3.1.1 Grid Convergence-------------------------------26
3.1.2 Periodic State---------------------------------26
3.1.3 Comparison of Lift Coefficient-----------------26
3.1.4 Domain Convergence-----------------------------26
3.2 Symmetric Rotating Motion------------------------27
3.3 Realistic Fruit Fly Motion-----------------------28
3.4 Overall Comparison-------------------------------28
4 Conclusion and Future Work-------------------------45
4.1 Conclusion---------------------------------------45
4.2 Future Work--------------------------------------46
Bibliography-----------------------------------------47

[1] C. S. Peskin, “Flow patterns around heart valves: a numerical method”, J. Comp.
Phys. 10: 252-271 (1972).
[2] C. S. Peskin, “The fluid dynamics of heart valves: Experimental, theoretical, and
computational methods”, Annu. Rev. Fluid Mech. 14: 235-259 (1982).
[3] R. Mittal, G. Iaccarino, “Immersed boundary method”, Annu. Rev. Fluid Mech. 37: 239-261 (2005)..
[4] J. Mohd-Yusof, “Combined immersed boundary/B-Spline method for simulations of flows in complex geometries in complex geometries CTR annual research briefs”, NASA Ames/Stanford University (1997).
[5] E. A. Fadlun, R. Verzicco, P. Orlandi, J. Mohd-Yusof, “Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations”, J. Comp. Phys. 161: 35-60 (2000).
[6] E. Balaras, “Modeling complex boundaries using an external force field on fixed
Cartesian grids in large-eddy simulations”, Comput. Fluids 33: 375-404 (2004).
[7] J. M. Yang, E. Balaras, “An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries”, J. Comp. Phys. 215: 12-40 (2006).
[8] J. Kim, D. Kim, H. Choi, “An immersed-boundary finite-volume method for simulations of flow in complex geometries”, J. Comp. Phys. 171: 132-150 (2001).
[9] D. Kim, H. Choi, “Immersed boundary method for flow around an arbitrarily moving body”, J. Comp. Phys. 212: 662-680 (2006).
[10] S. W. Su, M. C. Lai, C. A. Lin, “An immersed boundary technique for simulating complex flows with rigid boundary”, Comput. Fluids 36: 313-324 (2007).
[11] C. C. Liao, Y. W. Chang, C. A. Lin, J. M. McDonough, “Simulating flows with moving rigid boundary using immersed-boundary method”, Comput. Fluids 39: 152-167 (2010).
[12] P. H. Chang, C. C. Liao, H. W. Hsu, S. H. Liu, C. A. Lin, “Simulations of laminar and turbulent flows over periodic hills with immersed boundary method”, Comput. Fluids 92: 233-243 (2014).
[13] S. Balay, K. Buschelman, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. Mclnnes, PETSc Web page. (2010). (https://www.mcs.anl.gov/petsc)
[14] H. W. Hsu, F. N. Hwang, Z. H. Wei, S. H. Lai, C. A. Lin, “A parallel multilevel preconditioned iterative pressure Poisson solver for the large-eddy simulation of turbulent flow inside a duct”, Comput. Fluids 45: 138-146 (2011).
[15] M. N. Chang, “A parallel multilevel preconditioned iterative pressure Poisson solver for 3D lid-driven cavity”, Master thesis, Department of Mechanical Engineering, National Tsing Hua University (2013).
[16] W. Shyy, M. Berg, D. Ljunqvist, “Flapping and flexible wings for biological and micro air vehicles”, Prog. Aerosp. Sci. 35: 455-505 (1999).
[17] T. Nakata, H. Liu, Y. Tanaka, N. Nishihashi, X. Wang, A. Sato, “Aerodynamics of a bio-inspired flexible flapping-wing micro air vehicle”, Bioinspir. Biomim. 6 (2011).
[18] K. Matthew, K. Karl, W. Henry, A. Alexander, “Development of the nano hummingbird: a tailless flapping wing micro air vehicle”, 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition (2012).
[19] C. P. Ellington, “The novel aerodynamics of insect flight: applications to micro-air vehicles”, J. Exp. Biol. 202: 3439-3448 (1999).
[20] A. L. R. Thomas, G. K. Taylor, R. B. Srygley, R. L. Nudds, R. J. Bomphrey, “Dragonfly flight: free-flight and tethered flow visualizations reveal a diverse array of unsteady lift-generating mechanisms, controlled primarily via angle of attack”, J. Exp. Biol. 207: 4299-4323 (2004).
[21] R. J. Wood, “The first takeoff of a biologically inspired at-scale robotic insect”, IEEE Trans. Robot. 24: 341-347 (2008).
[22] K. Y. Ma, P. Chirarattananon, S. B. Fuller, R. J. Wood, “Controlled flight of a biologically inspired, insect-scale robot”, Science 340: 603-607 (2013).
[23] S. P. Sane, “The aerodynamics of insect flight” J. Exp. Biol. 206: 4191-4208 (2003).
[24] L. Zheng, T. Hedrick, R. Mittal, “A comparative study of the hovering efficiency of flapping and revolving wings”, Bioinspir. Biomim. 8 (2013).
[25] E. W. Hawkes, D. Lentink, “Fruit fly scale robots can hover longer with flapping wings than with spinning wings”, J. Royal Soc. Interface 13 (2016).
[26] S. A. Ansari, R. Zbikowski, K. Knowles, “Aerodynamic modeling of insect-like flapping flight for micro air vehicles”, Prog. Aerosp. Sci. 42: 129-172 (2006).
[27] M. H. Dickinson, K. G. Gotz, “Unsteady aerodynamic performance of model wings at low Reynolds-numbers”, J. Exp. Biol. 174: 45-64 (1993).
[28] A. P. Willmott, C. P. Ellington, “The mechanics of flight in the hawkmoth Manduca sexta. I. Kinematics of hovering and forward flight”, J. Exp. Biol. 200: 2705-2722 (1997).
[29] S. N. Fry, R. Sayaman, M. H. Dickinson, “The aerodynamics of free-flight maneuvers in Drosophila”, Science 300: 495-498 (2003).
[30] C. P. Ellington, C. vandenBerg, A. P. Willmott, A. L. R. Thomas, “Leading-edge vortices in insect flight”, Nature 384: 626-630 (1996).
[31] M. H. Dickinson, F. Lehmann, S. P. Sane, “Wing rotation and the aerodynamic basis of insect flight”, Science 284: 1954-1960 (1999).
[32] M. Sun, J. Tang, “Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion”, J. Exp. Biol. 205: 55-70 (2002).
[33] M. Sun, J. Tang, “Lift and power requirements of hovering flight in Drosophila virilis”, J. Exp. Biol. 205: 2413-2427 (2002).
[34] H. Aono, F. Liang, H. Liu, “Near- and far-field aerodynamics in insect hovering flight: an integrated computational study”, J. Exp. Biol. 211: 239-257 (2008).
[35] H. Liu, H. Aono, “Size effects on insect hovering aerodynamics: an integrated computational study”, Bioinspir. Biomim. 4 (2009).
[36] T. Weis-Fogh, “Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production”, J. Exp. Biol. 59: 169-230 (1973).
[37] F. O. Lehmann, S. P. Sane, M. Dickinson, “The aerodynamic effects of wing-wing interaction in flapping insect wings”, J. Exp. Biol. 208: 3075-3092 (2005).
[38] L. A. Miller, C. S. Peskin, “A computational fluid dynamics of ‘clap and fling’ in the smallest insects”, J. Exp. Biol. 208: 195-212 (2005).
[39] M. Sun, X. Yu, “Aerodynamic force generation in hovering flight in a tiny insect”, AIAA 44: 1532-1540 (2006).
[40] F. O. Lehmann, S. Pick, “The aerodynamic benefit of wing-wing interaction depends on stroke trajectory in flapping insect wings”, J. Exp. Biol. 210: 1362-1377 (2007).
[41] X. Yu, M. Sun, “A computational study of the wing-wing and wing-body interactions of a model insect”, Acta Mech. Sin. 25: 421-431 (2009).
[42] K. B. Lua, T. T. Lim, K. S. Yeo, “Scaling of aerodynamic forces of three-dimensional flapping wings”, AIAA 52: 1095-1101 (2014).