Skip to main content

Shallow and deep dynamic stall for flapping low Reynolds number airfoils

Experiments in Fluids volume 46, pages 883–901 (2009)Cite this article

Abstract

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Abbreviations

C L :

Airfoil lift coefficient per unit span

c :

Airfoil chord (= 152.4 mm)

f :

Airfoil oscillation pitch–plunge frequency

ω:

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

λ:

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

References

  • Anderson JM, Streitlien K, Barrett DS, Triantafyllou MS (1998) Oscillating foils of high propulsive efficiency. J Fluid Mech 360:41–72

    MATH  Article  MathSciNet  Google Scholar 

  • Carr L (1988) Progress in analysis and prediction of dynamic stall. J. Aircraft 25:6–17

    Article  Google Scholar 

  • Carr L, McCroskey WJ (1992) A review of recent advances in computational and experimental analysis of dynamic stall. In: IUTAM symposium on fluid dynamics of high angle of attack

  • Dickinson MH, Götz KG (1993) Unsteady aerodynamic performance of model wings at low Reynolds numbers. J Exp Biol 174:45–64

    Google Scholar 

  • Ellington CP (1984) The aerodynamics of hovering insect flight. I. The quasi-steady analysis. Philos Trans R Soc Lond B Biol Sci B 305:1–15

    Article  Google Scholar 

  • Choudhuri PG, Knight DD (1996) “Effects of compressibility, pitch rate, and Reynolds number on unsteady incipient leading-edge boundary layer separation over a pitching airfoil. J Fluid Mech 308:195–217

    MATH  Article  Google Scholar 

  • Jeon D (2000) Dissertation, Caltech

  • Jones KD, Dohring CM, Platzer MF (1996) Wake structures behind plunging airfoils: a comparison of numerical and experimental results. AIAA Paper 96-0078

  • Kamakoti R, Thakur S, Wright J, Shyy W (2006) Validation of a new parallel all-speed CFD code in a rule-based framework for multidisciplinary applications. AIAA-3063

  • Kang C-K, Baik YS, Bernal LP, Ol MV, Shyy W (2009) Fluid dynamics of pitching and plunging airfoils of Reynolds number between 1 × 10^4 and 6 × 10^4. AIAA-0536

  • Koochesfahani MM (1989) Vortical patterns in the wake of an oscillating airfoil. AIAA J. 27(9):1200–1205

    Article  Google Scholar 

  • Kramer B (2002) Experimental evaluation of superposition techniques applied to dynamic aerodynamics. AIAA-0700

  • Lehmann F-O (2004) The mechanisms of lift enhancement in insect flight. Naturwissenschaften 91:101–122

    Article  Google Scholar 

  • Leishman JG (2000) Principles of helicopter aerodynamics. Cambridge University Press, London

    Google Scholar 

  • Lian Y, Shyy W (2007) Aerodynamics of low reynolds number plunging airfoil under gusty environment. AIAA Paper-0071

  • Liiva J (1969) Unsteady aerodynamic and stall effects on helicopter rotor blade airfoil sections. J Aircraft 6(1):46–51

    Article  Google Scholar 

  • Liu H, Kawachi K (1998) A numerical study of insect flight. J Comput Phys 146:124–156

    MATH  Article  Google Scholar 

  • McCroskey WJ (1982) Unsteady airfoils. Ann Rev Fluid Mech 14:285–311

    Article  Google Scholar 

  • McCroskey WJ, Carr LW, McAlister KW (1976) Dynamic stall experiments on oscillating airfoils. AIAA J. 14(1):57–63

    Article  Google Scholar 

  • Menter FR (1993) Zonal two equation k-ω turbulence models for aerodynamic flows. AIAA-2906

  • Ohmi K, Coutanceau M, Loc TP, Dulieu A (1990) Vortex formation around an oscillating and translating airfoil at large incidences. J Fluid Mech 211:37–60

    Article  Google Scholar 

  • Ohmi K, Coutanceau M, Daube O, Loc TP (1991) Further experiments on vortex formation around an oscillating and translating airfoil at large incidences. J Fluid Mech 225:607–630

    Article  Google Scholar 

  • Ol M (2007) Vortical structures in high frequency pitch and plunge at low Reynolds number. AIAA-4233

  • Ol M, McAuliffe BR, Hanff ES, Scholz U, Kaehler Ch (2005) Comparison of laminar separation bubble measurements on a low Reynolds number airfoil in three facilities. AIAA-5149

  • Platzer M, Jones K (2006) Flapping wing aerodynamics—progress and challenges AIAA-0500

  • Poelma C, Dickinson MH (2006) Time-resolved reconstruction of the full velocity field around a dynamically-scaled flapping wing. Exp Fluids 41:213–225

    Article  Google Scholar 

  • Radespiel R, Windte J, Scholz U (2007) Numerical and experimental flow analysis of moving airfoils with laminar separation bubbles. AIAA J 45(6):1346–1356

    Article  Google Scholar 

  • Rao Y-J (1997) In-fibre bragg grating sensors. Meas Sci Tech 8(4):355–375

    Article  Google Scholar 

  • Shyy W (1985) A study of finite difference approximations to steady-state, convection-dominated flow problems. J Comput Phys 57(3):415–438

    MATH  Article  MathSciNet  Google Scholar 

  • Shyy W (1994) Computational modeling for fluid flow and interfacial transport. Elsevier, Amsterdam

    Google Scholar 

  • Shyy W, Udaykumar HS, Rao MM, Smith RW (2007) Computational fluid dynamics with moving boundaries. Taylor & Francis, Washington DC (1996, rev. printing 1997, 1998 & 2001); Dover, New York

  • Shyy W, Lian Y, Tang J, Viieru D, Liu H (2008) Aerodynamics of low Reynolds number flyers. Cambridge University Press,  

    Google Scholar 

  • Taylor GK, Nudds RL, Thomas ALR (2003) Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency. Nature (London) 425:707–711

    Article  Google Scholar 

  • Thomas PD, Lombard K (1978) The geometric conservation law—a link between finite-difference and finite-volume methods of flow computation on moving grids. AIAA-1208

  • Usherwood JR, Ellington CP (2002) The aerodynamics of revolving wings I. Model hawkmoth wings. J Exp Biol 205:1547–1564

    Google Scholar 

  • Visbal M, Shang JS (1989) Investigation of the flow structure around a rapidly pitching airfoil. AIAA J. 27(8):1044–1051

    Article  Google Scholar 

  • von Karman T, Sears WR (1938) Airfoil theory for nonuniform motion. J Aeronaut Sci 5(10):379–390

    Google Scholar 

  • Willert CE, Gharib M (1991) Digital particle image velocimetry. Exp Fluids 10(4)

  • Young J, Lai JCS (2004) Oscillation frequency and amplitude effects on the wake of a plunging airfoil. AIAA J 42(10):2042–2052

    Article  Google Scholar 

  • Zhang L, Zhang W, Bennion I (2002) In-fiber grating optic sensors. In: Yu FTS, Yin S (eds) Fiber optic sensors. Marcel Dekker, NY, pp 123–181

Download references

Author information

Authors and Affiliations

  1. Air Force Research Lab, Wright-Patterson AFB, Dayton, OH, USA

    Michael V. Ol

  2. Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, USA

    Luis Bernal, Chang-Kwon Kang & Wei Shyy

Authors
  1. Michael V. Ol
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luis Bernal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chang-Kwon Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wei Shyy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael V. Ol.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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). https://doi.org/10.1007/s00348-009-0660-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00348-009-0660-3

Keywords

  • Vorticity
  • Particle Image Velocimetry
  • RANS
  • Flow Separation
  • Lift Coefficient