Abstract
Global recurrence plots (GRPs) and windowed recurrence quantification analysis (WRQA) are two recurrence paradigms which find wide applications to detect the onset of instability in a dynamic system. The present work reports the attempt to employ these recurrence paradigms to assess the effect of frontal gust on the force patterns of an insect-sized flapping wing in the inclined-stroke plane. Horizontal and vertical forces generated by the flapping wing in the presence of gusts of the form \( \frac{{{\text{u}}_{\text{G}} }}{{{\text{u}}_{\text{w}} }} = \frac{{{\text{u}}_{\infty } }}{{{\text{u}}_{\text{w}} }} + \left( {\frac{{{\text{u}}_{\text{g}} }}{{{\text{u}}_{\text{w}} }}} \right)\sin \left( {2\uppi\frac{{{\text{f}}_{\text{g}} }}{{{\text{f}}_{\text{w}} }}{\text{t}}} \right) \) were numerically estimated in the 2D reference frame for Re = 150. Nine gusts with combinations of the ratio of gust frequency to wing’s flapping frequency, fg/fw = 0.1, 0.5 and 1 and ratio of gust velocity amplitude to root mean square averaged flapping velocity, ug/uw = 0.1, 0.5 and 1 were considered. Recurrence studies of the forces were carried out to find out the gusty condition, which would trigger an onset of unstable behaviour. Studies indicated a possible onset of instability in the force patterns for gust with fg/fw = 0.1 and ug/uw = 1. The onset of unstable behaviour was prominently captured by WRQA of the vertical force coefficient based on determinism (DET) and laminarity (LAM) series.
Similar content being viewed by others
Abbreviations
- c :
-
wing chord length, cm
- f w :
-
wing flapping frequency, Hz
- f* :
-
non-dimensionalized wing flapping frequency, \( \frac{1}{{2\uppi\left( {\frac{{{\text{A}}_{0} }}{\text{c}}} \right)}} \)
- l:
-
diagonal line
- l min :
-
minimum threshold diagonal line
- m:
-
dimensional phase space trajectory
- t :
-
time, sec
- t* :
-
non-dimensionalized time
- t w , T :
-
period of flapping in second
- u g :
-
gust amplitude, m/s
- u w :
-
root mean square average flapping velocity at the tip of the wing, m/s
- u Resultant :
-
resultant velocity, m/s
- u G :
-
gust velocity, m/s
- u ∞ :
-
mean free stream velocity, m/s
- \( {\vec{\text{u}}} \) :
-
flow velocity, m/s
- \( \overrightarrow {{{\text{u}}_{\text{g}} }} \) :
-
velocity of the moving mesh, m/s
- v:
-
length of vertical structures in recurrence plot
- vmin :
-
minimum threshold vertical line
- Ao :
-
stroke length of the wing, cm
- B:
-
pitching angle amplitude, deg
- CH :
-
coefficient of horizontal force
- CV :
-
coefficient of vertical force
- FDrag :
-
drag force, Newton
- FHorizontal :
-
horizontal force, Newton
- FLift :
-
lift force, Newton
- FResultant :
-
resultant force, Newton
- FVertical :
-
vertical force, Newton
- Lmax :
-
maximum diagonal structure of the recurrence plot
- N:
-
length of data series
- \( P^{\varepsilon } \left( l \right) \) :
-
frequency distribution of the diagonal lengths l
- \( P^{\varepsilon } \left( v \right) \) :
-
frequency distribution of vertical length, v
- \( R_{i,j}^{m,\varepsilon } \) :
-
recurrence matrix of an m-dimensional phase space trajectory and a neighbourhoods radius ε
- \( {\text{S}}_{\upphi} \) :
-
source term
- \( {\text{V}}\!\!\!\!\!- \) :
-
arbitrary control volume
- α(t):
-
instantaneous pitching angle, deg
- α0 :
-
mean pitching angle, deg
- β:
-
stroke plane angle, deg
- ϒ:
-
elliptical flow domain around the wing
- ε:
-
neighbourhood radius
- ø:
-
a scalar quantity
- ρ:
-
fluid density, kg/m3
- Γ:
-
diffusion coefficient
References
Ellington C P 1984 The Aerodynamics of hovering insect flight: III. Kinematics. Philos. Trans. R. Soc. London Ser. B 305: 41–78
Berman G and Wang Z J 2007 Energy-minimizing kinematics in hovering insect flight. J. Fluid Mech. 582: 153–168
Meng X and Sun M 2016 Wing kinematics, aerodynamic forces and vortex-wake structures in fruit-flies in forward flight. J. Bionic Eng. 13: 478–490
Ansari S A, Knowles K and Zbikowski R 2008 Insectlike flapping wings in the hover: Part II Effect of wing geometry. J. Aircraft 45: 1976–1990
Singh B and Chopra I 2008 Insect-based hover-capable flapping wings for micro air vehicles: experiments and analysis. AIAA J. 46: 2115–2135
Meng X, Liu Y and Sun M 2017 Aerodynamics of ascending flight in fruit flies. J. Bionic Eng. 14: 75–87
Berg A M and Biewerner A A 2008 Kinematics and power requirements of ascending and descending flight in the pigeon. J. Exp. Biol. 211: 1120–1130
Nagai H, Isogai K, Fujimoto T and Hayase T 2009 Experimental and numerical study of forward flight aerodynamics of insect flapping wing. AIAA J. 47: 730–742
Xiang J, Du J, Li D and Liu K 2016 Aerodynamic performance of the locust wing in gliding mode at low Reynolds number. J. Bionic Eng. 13: 249–260
Fry S N, Sayaman R and Dickenson M H 2003 The aerodynamics of free-flight maneuvers in drosophila. Science 300: 495–498
Broering T M and Lian Y 2012 The effect of phase angle and wing spacing on tandem flapping wings. Acta Mech. Sin. 28: 1557–1571
Combes S A and Daniel T L 2003 Flexural stiffness in insect wings I. Scaling and the influence of wing venation. J. Exp. Biol. 206: 2979–2987
Combes S A and Daniel T L 2003 Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending. J. Exp. Biol. 206: 2989–2997
Heathcote S and Gursul I 2007 Flexible flapping airfoil propulsion at low Reynolds number. AIAA J. 45: 1066–1079
Young J, Walker S M, Bomphery R J, Taylor G K and Thomas A L R 2009 Details of insect wing design and deformation enhance aerodynamic function and flight efficiency. Science 325: 1549–1552
Geng B, Xue Q, Zheng X, Liu G, Ren Y and Dong H 2017 The effect of wing flexibility on sound generation of flapping wings. Bioinspir. Biomim. 13. https://doi.org/10.1088/1748-3190/aa8447
Gao T and Lu X 2008 Insect normal hovering flight in ground effect. Phys. Fluids 20: 087101-1–11
Srinidhi N G and Vengadesan S 2017 Ground effect on tandem flapping wing hovering. Comput. Fluids 152: 40–56
Manoukis N C, Butail S, Diallo M, Ribeiro J M C and Paley D A 2014 Stereoscopic video analysis of Anopheles gambiae behavior in the field: Challenges and opportunity. Acta Trop. 132: S80–S85
Sane S P 2003 The aerodynamics of insect flight. J. Exp. Biol. 206: 4191–4208
Platzer M F, Jones K D, Young J and Lai J C S 2008 Flapping-wing aerodynamics: Progress and Challenges. AIAA J. 46: 2136–2155
Shyy W, Aono H, Chimakurthi S K, Trizila P, Kang C K, Cesnik C E S and Liu H 2010 Recent progress in flapping wing aerodynamics and aeroelasticity. Prog. Aerosp. Sci. 46: 284–327
Ward T A, Rezadad M, Fearday C J and Viyapuri R. A 2015 Review of Biomimetic air vehicle research: 1984–2014. Int. J. Micro Air Veh. 7: 375–394
Watkins S, Milbank J, Loxton B J and Melbourne W H 2006 Atmospheric Winds and Their Implications for Micro air Vehicles. AIAA J. 44: 2591–2600
Lian Y and Shyy W 2007 Aerodynamics of Low Reynolds Number Plunging Airfoil under Gusty Environment. In: 45th AIAA Aerospace Sciences Meeting and Exhibit. AIAA Paper 2007-70. pp 1–20
Wan T and Huang C 2008 Numerical Simulation of Flapping Wing Aerodynamic Performance under Gust Wind Conditions. In: 26th International Congress of the Aeronautical Sciences. pp. 1–11
Lian Y 2009 Numerical study of a flapping airfoil in gusty environments. In: 27th AIAA Applied Aerodynamics Conference. AIAA-2009-3952. pp. 1–13
Viswanath K and Tafti D K 2010 Effect of frontal gusts on forward flapping flight. AIAA J. 48: 2049–2062
Prater R and Lian Y 2012 Aerodynamic response of stationary and flapping wings in oscillatory low Reynolds number flows. In: 50th AIAA Aerospace Science Meeting including the New Horizons Forum and Aerospace Exposition. AIAA-2012-0418. pp. 1 – 17
Sarkar S, Chajjed S and Krishnan A 2013 Study of asymmetric hovering in flapping flight. Eur. J. Mech. B Fluids 37: 72–89
Zhu J, Jiang L, Zhao H, Tao B and Lei B 2015 Numerical study of a variable camber plunge airfoil under wind gust condition. J. Mech. Sci. Technol. 29: 4681–4690
Jones M and Yamaleev N K 2016 Effect of lateral, downward and frontal gusts on flapping wing performance. Comput. Fluids 140: 175–190
Srinidhi N G and Vengadesan S 2017 Lagrangian Coherent Structures in Tandem Flapping Wing Hovering. J. Bionic Eng. 14: 307–316
Durmaz O, Karaca H D, Ozen G D, Kasnakoglun and Kurtulus D F 2013 Dynamical modelling of the flow over a flapping wing using proper orthogonal decomposition and system identification techniques. Math. Comput. Model. Dyn. Syst. 19(2): 133–158
Marwan N 2008 A historical review of recurrence plots. Eur. Phys. J. Spec. Top. 164: 3–12
Poincaré H 1890 On the problem of three bodies and equations of dynamics. Acta Math. 13: 1–270
Monk A T and Compton A H 1939 Recurrence phenomena in cosmic-ray intensity. Rev. Mod. Phys. 11(3–4): 173–179
Eckmann J P, Kamphorst S O and Ruelle D 1987 Recurrence plots of dynamical systems. Europhys. Lett. 4: 973–977
Zbilut J P, Giuliani A and Webber C L Jr 1998 Recurrence quantification analysis and principal components in the detection of short complex signals. Phys. Lett. A. 237: 131–135
Iwanski J S and Bradley E 1998 Recurrence plots of experimental data: to embed or not to embed? Chaos 8: 861–871
Choi J M, Bae B H and Kim S Y 1999 Divergence in perpendicular recurrence plot: quantification of dynamical divergence from short chaotic time series. Phys. Lett. A 263: 299–306
Horai S, Yamada T and Aihara K 1996 Determinism analysis with iso-directional recurrence plots. IEEE Trans. Inst. Electric. Eng. Jpn. C 122: 141–147
Manuca R and Savit R 1996 Stationarity and non-stationarity in time series analysis. Physica D 99: 134–161
Casdagli M C 1997 Recurrence plots revisited. Physica D 108: 12–44
Zbilut J P and Webber C L Jr 1992 Embeddings and delays as derived from quantification of recurrence plots. Phys. Lett. A 171: 199–203
Marwan N, Wessel N, Meyerfeldt U, Schirdewan A and Kurths J 2002 Recurrence plot based measures of complexity and its application to heart rate variability data. Phys. Rev. E 66: 026702
Badrinath S, Bose C and Sarkar S 2017 Identifying the route to chaos in the flow past a flapping airfoil. Eur. J. Mech. B/ Fluids 66: 38–59
Bose C, Reddy V, Gupta S and Sarkar S 2017 Transient and Stable Chaos in Dipteran Flight Inspired Flapping Motion. J. Comput. Nonlin. Dyn. 13: 021014
Bos F M, Lentink D, Oudheusden B W V and Bijl H 2008 Influence of wing kinematics on aerodynamic performance in hovering insect flight. J. Fluid Mech. 594: 341–368
Wood R J, Finio B, Karpelson M and Whitney J P 2012 Progress on pico air vehicles. Int. J. Robot. Res. 31: 1292–1302
Brodsky A K 1994 The Evolution of Insect Flight, Oxford: Oxford University Press
Henderson R D 1995 Details of the drag curve near the onset of vortex shedding. Phys. Fluids 7: 2102–2104
Williamson C H K 1995 Book Chapter: Vortex dynamics in the wake of a cylinder, Fluid Vortices. SI edition, Amsterdam, Holland, Kluwer Academic Publishing, pp. 155–234
Ferziger J H and Peric M 2002 Computational Methods for Fluid Dynamics. 3rd Edition, Heidelberg New York: Springer-Verlag Berlin
Issa R I 1985 Solution of the implicitly discretized fluid flow equations by operator-splitting. J. Comput. Phys. 65: 40–65
Wang Z J 2000 Two dimensional mechanism for insect hovering. Phys. Rev. Lett. 85: 2216–2219
Sudhakar Y and Vengadesan S 2010 Flight force production by flapping insect wings in inclined-stroke plane. Comput. Fluids 39: 683–695
Xu S and Wang Z J 2006 An immersed interface method for simulating the interaction of a fluid with moving boundaries. J. Comput. Phys. 216: 454–493
Harland C and Jacob J D 2010 Gust load testing in a low-cost MAV gust and shear tunnel. In: 27th AIAA Aerodynamic Measurement Technology and Ground Testing Conference. AIAA-2010-4539. pp. 1–14
Zbilut J P, Zaldvar C J M and Strozzi F 2002 Recurrence quantification based Liapunov exponents for monitoring divergence in experimental data. Phys. Lett. A 297: 173–181
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
DE MANABENDRA, M., MATHUR, J.S. & VENGADESAN, S. Recurrence studies of insect-sized flapping wings in inclined-stroke plane under gusty conditions. Sādhanā 44, 67 (2019). https://doi.org/10.1007/s12046-018-1036-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12046-018-1036-2