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Theoretical and Computational Fluid Dynamics

, Volume 27, Issue 6, pp 843–864 | Cite as

An unsteady airfoil theory applied to pitching motions validated against experiment and computation

  • Kiran RameshEmail author
  • Ashok Gopalarathnam
  • Jack R. Edwards
  • Michael V. Ol
  • Kenneth Granlund
Original Article

Abstract

An inviscid theoretical method that is applicable to non-periodic motions and that accounts for large amplitudes and non-planar wakes (large-angle unsteady thin airfoil theory) is developed. A pitch-up, hold, pitch-down motion for a flat plate at Reynolds number 10,000 is studied using this theoretical method and also using computational (immersed boundary method) and experimental (water tunnel) methods. Results from theory are compared against those from computation and experiment which are also compared with each other. The variation of circulatory and apparent-mass loads as a function of pivot location for this motion is examined. The flow phenomena leading up to leading-edge vortex shedding and the limit of validity of the inviscid theory in the face of vortex-dominated flows are investigated. Also, the effect of pitch amplitude on leading-edge vortex shedding is examined, and two distinctly different vortex-dominated flows are studied using dye flow visualizations from experiment and vorticity plots from computation.

Keywords

Unsteady aerodynamics Airfoil aerodynamics Leading-edge vortex 

Nomenclature

α

Angle between the airfoil and inertial horizontal

Δp

Pressure difference over airfoil

\({\dot{\alpha}}\)

Pitch rate

\({\dot{h}}\)

Plunge rate

η(x)

Variation of camber along airfoil

γ(θ, t)

Chordwise distribution of bound vorticity on airfoil

Γ(i)

Bound circulation of airfoil at time i

\({\Gamma_{w_{k}}}\)

Strength of wake vortex shed at time k − 1

\({\phi}\)

Velocity potential

\({\phi_{B}}\)

Velocity potential from bound circulation

\({\phi_{W}}\)

Velocity potential from wake circulation

ρ

Air density

θ

Variable of transformation of chordwise distance

\({\omega}\)

Rate of rotation of the body frame

n

Unit vector normal to camberline in body frame

r

Position vector of a point in the body frame

V0

Velocity of the body frame with respect to the inertial frame

a

Pivot location on the airfoil from 0 to 1 (x/c)

A0, A1, A2....

Fourier coefficients

Bxyz

Body frame

c

Airfoil chord

Cd

Drag coefficient

Cl

Lift coefficient

CN

Normal force coefficient

CS

Leading edge suction force coefficient

FN

Normal force on airfoil

FS

Leading edge suction force

h

Plunge displacement in the inertial Z direction

K

Non-dimensional pitch rate = \({\dot{\alpha}c/(2U)}\)

L

Lift

OXY Z

Inertial frame

p

Pressure distribution over airfoil

p

Freestream pressure

t

Time

t*

Non-dimensional time = tU/c

U

Freestream velocity

W

Local downwash

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Kiran Ramesh
    • 1
    Email author
  • Ashok Gopalarathnam
    • 1
  • Jack R. Edwards
    • 1
  • Michael V. Ol
    • 2
  • Kenneth Granlund
    • 2
  1. 1.Department of Mechanical and Aerospace EngineeringNorth Carolina State UniversityRaleighUSA
  2. 2.U.S. Air Force Research LaboratoryAir Vehicles Directorate, WPAFBDaytonUSA

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