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Shallow and deep dynamic stall for flapping low Reynolds number airfoils


We consider a combined experimental (based on flow visualization, direct force measurement and phase-averaged 2D particle image velocimetry in a water tunnel), computational (2D Reynolds-averaged Navier–Stokes) and theoretical (Theodorsen’s formula) approach to study the fluid physics of rigid-airfoil pitch–plunge in nominally two-dimensional conditions. Shallow-stall (combined pitch–plunge) and deep-stall (pure-plunge) are compared at a reduced frequency commensurate with flapping-flight in cruise in nature. Objectives include assessment of how well attached-flow theory can predict lift coefficient even in the presence of significant separation, and how well 2D velocimetry and 2D computation can mutually validate one another. The shallow-stall case shows promising agreement between computation and experiment, while in the deep-stall case, the computation’s prediction of flow separation lags that of the experiment, but eventually evinces qualitatively similar leading edge vortex size. Dye injection was found to give good qualitative match with particle image velocimetry in describing leading edge vortex formation and return to flow reattachment, and also gave evidence of strong spanwise growth of flow separation after leading-edge vortex formation. Reynolds number effects, in the range of 10,000–60,000, were found to influence the size of laminar separation in those phases of motion where instantaneous angle of attack was well below stall, but have limited effect on post-stall flowfield behavior. Discrepancy in lift coefficient time history between experiment, theory and computation was mutually comparable, with no clear failure of Theodorsen’s formula. This is surprising and encouraging, especially for the deep-stall case, because the theory’s assumptions are clearly violated, while its prediction of lift coefficient remains useful for capturing general trends.

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C L :

Airfoil lift coefficient per unit span

c :

Airfoil chord (= 152.4 mm)

f :

Airfoil oscillation pitch–plunge frequency


Pitch–plunge circular frequency

h :

Plunge position as function of time

h 0 :

Nondimensional plunge amplitude

k :

Reduced frequency of pitch or plunge, k = ωc/2U ∞:  = πfc/U

St :

Strouhal number, 2fch 0/U ∞:  = 0.08

U :

Freestream (reference) velocity (cm/s)

Re :

Reynolds number, Re = c U /ν, ν taken as 0.93 × 10−6 in SI units for water at 23°C

α :

Kinematic angle of incidence due to pitch

A :

Pitch amplitude (in degrees)

α 0 :

Mean angle of attack (i.e., the constant pitch angle offset from zero)

α e :

Total angle of attack from trigonometric combination of pitch and plunge

x p :

Pitch pivot point: fraction of chord downstream from airfoil leading edge (= 0.25)

t/T :

Dimensionless time, in fractions of one oscillation period

φ :

Phase difference between pitching and plunging; positive → pitch leads


Ratio of pitch-amplitude to plunge-induced angle of attack


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Correspondence to Michael V. Ol.

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Ol, M.V., Bernal, L., Kang, CK. et al. Shallow and deep dynamic stall for flapping low Reynolds number airfoils. Exp Fluids 46, 883–901 (2009).

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