Abstract
Numerical simulations are conducted to investigate the mechanism of hovering flight with an inclined stroke. The Reynolds numbers considered are 150 and 1000 based on the maximum translational velocity and wing chord length. Three mechanisms responsible for high vertical force generation, suggested by Dickinson et al. (1999), are confirmed and more elaborated in the present study. First, we show that the vertical force during downstroke is larger than that from the quasi-steady analysis due to the delayed stall mechanism. Second, the wing-wake interaction of reducing the negative vertical force during the stroke reversal is explained in terms of the reattachment of the vortex, shed previously during downstroke, on the wing, by which the wing is submerged in a low pressure region during upstroke and has a smaller negative vertical force. Finally, the rotational circulation is explained by advancing the rotation timing of the wing at supination using both the numerical simulation and inviscid potential theory.
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Abbreviations
- a :
-
Major axis of the ellipse
- A m :
-
Stroke amplitude
- b :
-
Minor axis of the ellipse
- c :
-
Chord length of the wing
- C D :
-
Drag coefficient
- C H :
-
Horizontal force coefficient
- C L :
-
Lift coefficient
- C rot :
-
Hydrodynamic lift force coefficient due to wing rotation
- C V :
-
Vertical force coefficient
- C Va :
-
Vertical force coefficient due to added mass
- C Vqs :
-
Vertical force coefficient with the quasisteady assumption
- f :
-
Wing beat frequency
- g :
-
Gravity acceleration
- h :
-
Thickness of the wing
- l :
-
Wing span
- p :
-
Pressure
- Re :
-
Reynolds number,Re =u max c/v
- t :
-
Time
- T :
-
Period of the flapping motion
- u d ,u d :
-
Translational velocity of the wing
- umax :
-
Maximum translational velocity during flapping
- x′:
-
Position vector in a moving reference frame, (x′,y′)
- (x c ,y c ):
-
Position of the wing center
- α:
-
Angle between the chord axis and the stroke plane
- α d :
-
Constant value of α during the down-stroke period
- ga d :
-
Optimal αd for maximum vertical force
- αu :
-
Constant value of α during the upstroke period
- β:
-
Angle of the stroke plane to the horizontal axis
- v :
-
Kinematic viscosity
- ρ:
-
Fluid density
- Ω, ω:
-
Angular velocity of the wing
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Kim, D., Choi, H. Two-dimensional mechanism of hovering flight by single flapping wing. J Mech Sci Technol 21, 207–221 (2007). https://doi.org/10.1007/BF03161726
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DOI: https://doi.org/10.1007/BF03161726