Abstract
A leading-edge suction parameter (LESP) that is derived from potential flow theory as a measure of suction at the airfoil leading edge is used to study initiation of leading-edge vortex (LEV) formation in this article. The LESP hypothesis is presented, which states that LEV formation in unsteady flows for specified airfoil shape and Reynolds number occurs at a critical constant value of LESP, regardless of motion kinematics. This hypothesis is tested and validated against a large set of data from CFD and experimental studies of flows with LEV formation. The hypothesis is seen to hold except in cases with slow-rate kinematics which evince significant trailing-edge separation (which refers here to separation leading to reversed flow on the aft portion of the upper surface), thereby establishing the envelope of validity. The implication is that the critical LESP value for an airfoil–Reynolds number combination may be calibrated using CFD or experiment for just one motion and then employed to predict LEV initiation for any other (fast-rate) motion. It is also shown that the LESP concept may be used in an inverse mode to generate motion kinematics that would either prevent LEV formation or trigger the same as per aerodynamic requirements.
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Abbreviations
- \(\alpha \) :
-
Angle between the airfoil and inertial horizontal
- \(\dot{\alpha }\) :
-
Pitch rate
- \(\dot{h}\) :
-
Plunge rate
- \(\eta \) :
-
Variation of camber along airfoil
- \(\gamma \) :
-
Chordwise distribution of bound vorticity on airfoil
- \(\varGamma _b\) :
-
Bound circulation of airfoil at time t
- \(\varGamma _{\mathrm{tev}_m}\) :
-
Strength of mth wake/trailing-edge vortex
- \(\omega \) :
-
Angular frequency
- \(\phi \) :
-
Velocity potential
- \(\phi _\mathrm{B}\) :
-
Velocity potential from bound circulation
- \(\phi _\mathrm{tev}\) :
-
Velocity potential from trailing-edge vorticity (wake circulation)
- \(\theta \) :
-
Variable of transformation of chordwise distance
- \(A_{0},A_{1},A_{2}\ldots \) :
-
Fourier coefficients
- Bxyz :
-
Body frame
- c :
-
Airfoil chord
- \(C_\mathrm{f}\) :
-
Skin friction coefficient
- \(C_\mathrm{p}\) :
-
Pressure coefficient
- h :
-
Plunge displacement in the inertial Z direction
- \(K = \dot{\alpha } c/2U\) :
-
Reduced frequency (ramp)
- \(k = \omega c/2U\) :
-
Reduced frequency (sinusoidal)
- LESP:
-
Leading-edge suction parameter
- \(\hbox {LESP}_\mathrm{crit}\) :
-
Critical value of LESP corresponding to LEV initiation
- OXYZ :
-
Inertial frame
- Re :
-
Reynolds number
- t :
-
Time
- \(t^*\) :
-
Nondimensional time \(=tU/c\)
- U :
-
Freestream velocity
- W :
-
Local downwash
- \(x_\mathrm{p}\) :
-
Pivot location on the airfoil from 0 to c
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Communicated by Jeff D. Eldredge.
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Ramesh, K., Granlund, K., Ol, M.V. et al. Leading-edge flow criticality as a governing factor in leading-edge vortex initiation in unsteady airfoil flows. Theor. Comput. Fluid Dyn. 32, 109–136 (2018). https://doi.org/10.1007/s00162-017-0442-0
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DOI: https://doi.org/10.1007/s00162-017-0442-0