Acta Mechanica Sinica

, Volume 34, Issue 5, pp 839–854 | Cite as

Lagrangian-based numerical investigation of aerodynamic performance of an oscillating foil

  • Mengjie Zhang
  • Qin WuEmail author
  • Biao Huang
  • Guoyu Wang
Research Paper


The dynamic stall problem for blades is related to the general performance of wind turbines, where a varying flow field is introduced with a rapid change of the effective angle of attack (AOA). The objective of this work is to study the aerodynamic performance of a sinusoidally oscillating NACA0012 airfoil. The coupled \(k{-}\omega \) Menter’s shear stress transport (SST) turbulence model and \(\gamma {-}Re_{\uptheta }\) transition model were used for turbulence closure. Lagrangian coherent structures (LCS) were utilized to analyze the dynamic behavior of the flow structures. The computational results were supported by the experiments. The results indicated that this numerical method can well describe the dynamic stall process. For the case with reduced frequency \(K = 0.1\), the lift and drag coefficients increase constantly with increasing angle prior to dynamic stall. When the AOA reaches the stall angle, the lift and drag coefficients decline suddenly due to the interplay between the first leading- and trailing-edge vortex. With further increase of the AOA, both the lift and drag coefficients experience a secondary rise and fall process because of formation and shedding of the secondary vortex. The results also reveal that the dynamic behavior of the flow structures can be effectively identified using the finite-time Lyapunov exponent (FTLE) field. The influence of the reduced frequency on the flow structures and energy extraction efficiency in the dynamic stall process is further discussed. When the reduced frequency increases, the dynamic stall is delayed and the total energy extraction efficiency is enhanced. With \(K = 0.05\), the amplitude of the dynamic coefficients fluctuates more significantly in the poststall process than in the case of \(K = 0.1\).


Oscillating foil Dynamic stall Lagrangian coherent structures Computational fluid dynamics (CFD) 

List of symbols

\(\mu \)

Dynamic viscosity \((\hbox {kg}/\hbox {m} {\cdot } \hbox {s})\)


Velocity in the i-th direction (m/s)


Position (m)


Turbulent kinetic energy \((\hbox {m}^{2}/\hbox {s}^{2})\)

\(\omega \)

Specific turbulent dissipation rate (1/s)


Reynolds number


Angular velocity (1/s)

\(\delta \)

Lyapunov exponent

\(\omega _{z}\)

z-vorticity (1/s)


Second invariant of velocity gradient tensor (\(1/\hbox {s}^{2}\))

\(\Delta {t}\)

Time step (s)


Torque \((\hbox {N}{\cdot }\hbox {m})\)


Time period of one cycle (s)


Pitching angular velocity (1/s)

\(\rho \)

Density (\(\hbox {kg/m}^{3}\))



This work was supported by the National Postdoctoral Program for Innovative Talents (Grant BX201700126), the China Postdoctoral Science Foundation (Grant 2017M620043), the National Natural Science Foundation of China (Grants 51679005 and 91752105), and the National Natural Science Foundation of Beijing (Grant 3172029).


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Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Mengjie Zhang
    • 1
  • Qin Wu
    • 1
    • 2
    Email author
  • Biao Huang
    • 1
  • Guoyu Wang
    • 1
  1. 1.School of Mechanical EngineeringBeijing Institute of TechnologyBeijingChina
  2. 2.Department of Thermal EngineeringTsinghua UniversityBeijingChina

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