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Recurrence studies of insect-sized flapping wings in inclined-stroke plane under gusty conditions


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.

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c :

wing chord length, cm

f w :

wing flapping frequency, Hz

f* :

non-dimensionalized wing flapping frequency, \( \frac{1}{{2\uppi\left( {\frac{{{\text{A}}_{0} }}{\text{c}}} \right)}} \)


diagonal line

l min :

minimum threshold diagonal line


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


length of vertical structures in recurrence plot

vmin :

minimum threshold vertical line

Ao :

stroke length of the wing, cm


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


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


instantaneous pitching angle, deg

α0 :

mean pitching angle, deg


stroke plane angle, deg


elliptical flow domain around the wing


neighbourhood radius


a scalar quantity


fluid density, kg/m3


diffusion coefficient


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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).

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  • Insect-sized flapping wing
  • inclined-stroke plane
  • frontal gust
  • global recurrence plots
  • windowed recurrence quantification analysis