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

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.

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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

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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). https://doi.org/10.1007/s12046-018-1036-2

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  • DOI: https://doi.org/10.1007/s12046-018-1036-2

Keywords

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