A flight mechanics-based justification of the unique range of Strouhal numbers for avian cruising flight

Abstract

In this paper, analytical expressions for cycle-averaged aerodynamic forces generated by flapping wings are derived using a force model and flapping kinematics suitable for the forward flight of avian creatures. A strip theory-based formulation is proposed and the analytical expressions are found as functions of the amplitude of twist profile, mean twist angle, the flow separation point on the upper surfaces of the wings, and Strouhal number. Numerical results are obtained for a rectangular planform as well as for a representative avian wing planform. Utilizing these results, it is shown that there exists a narrow Strouhal number range where cycle-averaged net thrust, lift, and lift to drag ratio are optimal for a given flow pattern over the upper surfaces of the wings. This narrow Strouhal number range, found to be between 0.1 and 0.3, corresponds to the cruising range for a large number of avian creatures, as documented in current literature. An explanation, based on force constraints and local optimization in aerodynamic force generation, is provided for the unique range of Strouhal numbers utilized in avian cruising flight. The results and the approach outlined in the paper can be utilized to design efficient bio-inspired flapping vehicles.

Keywords

Avian-scale flapping aerodynamics flight mechanics

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References

Triantafyllou

Gopalkrishnan

Wake mechanics for thrust generation in oscillating foils. Phys of Fluids A: Fluid Dyn 1991; 3: 2835–2837.

Triantafyllou

Grosenbaugh

Optimal thrust development in oscillating foils with application to fish propulsion. J Fluids Struct 1993; 7: 205–224.

Anderson

Streitlien

Barrett

, et al. Oscillating foils of high propulsive efficiency. J Fluid Mech 1998; 360: 41–72.

Wang

Vortex shedding and frequency selection in flapping flight. J Fluid Mech 2000; 410: 323–341.

Taylor

Nudds

Thomas

Flying and swimming animals cruise at a strouhal number tuned for high power efficiency. Nature 2003; 425: 707–711.

Ellington

Insects versus birds: the great divide (invited). In: 44th AIAA aerospace sciences meeting and exhibit, Reno, Nevada, 9–12 January 2006.

Paranjape

Chung

Hilton

Optimizing the forces and propulsive efficiency in bird-scale flapping flight. In: AIAA atmospheric flight mechanics (AFM) conference, Boston, MA, 19–22 August 2013, paper no. AIAA 2013-4916.

Goman

Khrabrov

State-space representation of aerodynamic characteristics of an aircraft at high angles of attack. J Aircraft 1994; 31: 1109–1115.

Paranjape

Chung

Hilton

, et al. Dynamics and performance of tailless micro aerial vehicle with flexible articulated wings. AIAA J 2012; 50: 1177–1188.

10.

Withers

An aerodynamic analysis of bird wings as fixed aerofoils. J Exp Biol 1981; 90: 143–162.

11.

Betteridge

Archer

A study of the mechanics of flapping wings. Aeronaut Q 1974; 25: 129–142.

12.

Phlips

East

Pratt

An unsteady lifting line theory of flapping wings with application to the forward flight of birds. J Fluid Mech 1981; 112: 97–125.

13.

Theodorsen

General theory of aerodynamic instability and the mechanism of flutter. NACA Technical Report NACA-TR-496 1935; 496.

14.

Garrick

Propulsion of a flapping and oscillating airfoil. NACA Technical Report NACA-TR-567 1937; 567.

15.

Rayner

A vortex theory of animal flight. part 1. the vortex wake of a hovering animal. J Fluid Mech 1979; 91: 697–730.

16.

Rayner

A vortex theory of animal flight. part 2. the forward flight of birds. J Fluid Mech 1979; 91: 731–763.

17.

DeLaurier

An aerodynamic model for flapping-wing flight. Aeronaut J 1993; 97: 125–130.

18.

Bhattacharjee

Subbarao

Closed-form expressions for cycle-averaged aerodynamic quantities at an airfoil section of an avian flapping wing. In: AIAA scitech 2019 forum, San Diego, 17–19 January 2019, paper no. AIAA 2019-0565.

19.

Nakata

Liu

Bomphrey

A cfd-informed quasi-steady model of flapping-wing aerodynamics. J Fluid Mech 2015; 783: 323–343.

20.

Chin

Lentink

Flapping wing aerodynamics: from insects to vertebrates. J Exp Biol 2016; 219: 920–932.

21.

Orlowski

Girard

Modeling and simulation of nonlinear dynamics of flapping wing micro air vehicles. AIAA J 2011; 49: 969–981.

22.

Tuncer

Platzer

MF.

Computational study of flapping airfoil aerodynamics. J Aircraft 2000; 37: 514–520.

23.

Hall

Pigott

Hall

Power requirements for large-amplitude flapping flight. J Aircraft 1998; 35: 352–361.

24.

Hoerner

Fluid-dynamic drag. Brick Town, NJ: Hoerner Fluid Dynamics, 1965.

25.

Viieru

Albertani

Shyy

, et al. Effect of tip vortex on wing aerodynamics of micro air vehicles. J Aircraft 2005; 42: 1530–1536.

26.

Paranjape

Dorothy

Chung

, et al. A flight mechanics-centric review of bird-scale flapping flight. Int J Aeronaut Space Sci 2012; 13: 267–281.

27.

Tuncer

Kaya

Optimization of flapping airfoils for maximum thrust and propulsive efficiency. AIAA J 2005; 43: 2329–2336.

28.

Isogai

Shinmoto

Watanabe

Effects of dynamic stall on propulsive efficiency and thrust of flapping airfoil. AIAA J 1999; 37: 1145–1151.

29.

Heathcote

Wang

Gursul

Effect of spanwise flexibility on flapping wing propulsion. J Fluids Struct 2008; 24: 183–199.

30.

Doman

Oppenheimer

Sigthorsson

Dynamics and control of a minimally actuated biomimetic vehicle: part I – aerodynamic model. In: AIAA guidance, navigation, and control conference, Chicago, IL, 10–13 August 2013, paper no. AIAA 2009-6160.

31.

Orlowski

Girard

Longitudinal flight dynamics of flapping-wing micro air vehicles. J Guid, Control Dynmics 2012; 35: 1115–1131.

32.

Bhattacharjee

Paranjape

Pant

Optimization of the spanwise twist of a flapping wing for bird-sized aircraft using a quasi-steady aerodynamic model. Int J Aeronaut Space Sci 2019; 20: 571–583.

33.

Cleaver

Wang

Gursul

, et al. Lift enhancement by means of small-amplitude airfoil oscillations at low Reynolds numbers. AIAA J 2011; 49: 2018–2033.

34.

Oehme

Kitzler

On the geometry of the avian wing (studies on the biophysics and physiology of avian flight ii. Technical report, NASA-TT-F-16901, 1975.

35.

Tobalske

Dial

Flight kinematics of black-billed magpies and pigeons over a wide range of speeds. J Exp Biol 1996; 199: 263–280.

36.

Parslew

Crowther

Simulating avian wingbeat kinematics. J Biomech 2010; 43: 3191–3198.