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Effect of pitch on the flow behavior around a hovering wing


The effect of pitch on the evolution of flow and aerodynamic forces around a hovering flat-plate wing has been experimentally investigated in this study using particle image velocimetry and direct force measurements. The measurements are conducted on a wing at a reduced frequency of \(k=0.32\) and Reynolds number of \(Re=220\). The Lagrangian finite-time Lyapunov exponent method is used to analyze the unsteady flowfields by identifying dynamically relevant flow features and their evolution. First, the effect of a change in the duration of pitch for a symmetric pitch is discussed. The flow stages based on the LEV emergence, growth, lift-off, and decay remain the same for the compared cases whereas the duration of flow stages varies. Second, we introduce a phase lead and lag with respect to the stroke timing and detailed flow development is discussed for these cases. This is further corroborated with the measured aerodynamic forces to highlight the effect of varying the phase-shift on the different characteristics of the hovering wing. Changing the pitching phase results in distinct flow changes that correlate with a higher lift production when the pitch precedes the stroke reversal and lower lift production when the pitch succeeds the stroke reversal.

Graphical abstract

A schematic of differences in flow development based on the vorticity during advanced, symmetric, and delayed wing pitch. Blue features are indicative of the LEV and orange are indicative of the TEV. Maroon arrows indicate the magnitude and timing of the maximum lift in the half-stroke. Black arrows indicate the magnitude and timing of the maximum drag coefficients for three representative cases.

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\(\alpha \) :

Angle of attack (\(^{\circ }\))

\(\beta \) :

Pitch angle with respect to vertical (\(^{\circ }\))

\(\phi \) :

Stroke angle (\(^{\circ }\))

\({\dot{\phi }}\) :

Stroke velocity (\(\hbox {m s}^{-1}\))

\({\dot{\phi }}_\text {max}\) :

Maximum stroke velocity (\(\hbox {m s}^{-1}\))

\(\omega \) :

Vorticity (\(\hbox {s}^{-1}\))

c :

Chord length (m)

\(\left(c_\text {rp}\right)\) :

Distance of pitch axis to the leading edge

\(C_l, C_d\) :

Lift, drag coefficients

f :

Flapping frequency (Hz)

k :

Reduced frequency

R :

Wing span (m)

Re :

Reynolds number

s :

Saddle distance (m)

t :

Physical time (s)

\(t^*\) :

Convective time

\(t_1\) :

Integration time (s)

T :

Period of a flapping cycle (s)

\(T_\text {f}\) :

Pitch duration (s)

\(U^*= 2 {\hat{\phi }} f R\) :

Mean flapping velocity (\(\hbox {m s}^{-1}\))

\(u_{\phi }\) :

Induced velocity due to stroke (\(\hbox {m s}^{-1}\))

\(v_\text {t}\) :

Tangential velocity (\(\hbox {m s}^{-1}\))


Spatial coordinates (m)

\(\nabla \) :

Gradient function


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Correspondence to Swathi Krishna.

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Krishna, S., Green, M.A. & Mulleners, K. Effect of pitch on the flow behavior around a hovering wing. Exp Fluids 60, 86 (2019).

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