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 insectsized flapping wing in the inclinedstroke 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, f_{g}/f_{w} = 0.1, 0.5 and 1 and ratio of gust velocity amplitude to root mean square averaged flapping velocity, u_{g}/u_{w} = 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 f_{g}/f_{w} = 0.1 and u_{g}/u_{w} = 1. The onset of unstable behaviour was prominently captured by WRQA of the vertical force coefficient based on determinism (DET) and laminarity (LAM) series.
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Abbreviations
 c :

wing chord length, cm
 f _{ w } :

wing flapping frequency, Hz
 f* :

nondimensionalized 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* :

nondimensionalized 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
 v_{min} :

minimum threshold vertical line
 Ao :

stroke length of the wing, cm
 B:

pitching angle amplitude, deg
 C_{H} :

coefficient of horizontal force
 C_{V} :

coefficient of vertical force
 F_{Drag} :

drag force, Newton
 F_{Horizontal} :

horizontal force, Newton
 F_{Lift} :

lift force, Newton
 F_{Resultant} :

resultant force, Newton
 F_{Vertical} :

vertical force, Newton
 L_{max} :

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 mdimensional phase space trajectory and a neighbourhoods radius ε
 \( {\text{S}}_{\upphi} \) :

source term
 \( {\text{V}}\!\!\!\!\! \) :

arbitrary control volume
 α(t):

instantaneous pitching angle, deg
 α_{0} :

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/m^{3}
 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 Energyminimizing kinematics in hovering insect flight. J. Fluid Mech. 582: 153–168
Meng X and Sun M 2016 Wing kinematics, aerodynamic forces and vortexwake structures in fruitflies 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 Insectbased hovercapable 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 freeflight 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/10.1088/17483190/aa8447
Gao T and Lu X 2008 Insect normal hovering flight in ground effect. Phys. Fluids 20: 0871011–11
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 Flappingwing 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 200770. 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. AIAA20093952. 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. AIAA20120418. 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 cosmicray 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 isodirectional recurrence plots. IEEE Trans. Inst. Electric. Eng. Jpn. C 122: 141–147
Manuca R and Savit R 1996 Stationarity and nonstationarity 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: SpringerVerlag Berlin
Issa R I 1985 Solution of the implicitly discretized fluid flow equations by operatorsplitting. 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 inclinedstroke 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 lowcost MAV gust and shear tunnel. In: 27th AIAA Aerodynamic Measurement Technology and Ground Testing Conference. AIAA20104539. 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
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DE MANABENDRA, M., MATHUR, J.S. & VENGADESAN, S. Recurrence studies of insectsized flapping wings in inclinedstroke plane under gusty conditions. Sādhanā 44, 67 (2019). https://doi.org/10.1007/s1204601810362
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DOI: https://doi.org/10.1007/s1204601810362
Keywords
 Insectsized flapping wing
 inclinedstroke plane
 frontal gust
 global recurrence plots
 windowed recurrence quantification analysis