Skip to main content
Log in

Lagrangian-based investigation of the transient flow structures around a pitching hydrofoil

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

The objective of this paper is to address the transient flow structures around a pitching hydrofoil by combining physical and numerical studies. In order to predict the dynamic behavior of the flow structure effectively, the Lagrangian coherent structures (LCS) defined by the ridges of the finite-time Lyapunov exponent (FTLE) are utilized under the framework of Navier–Stokes flow computations. In the numerical simulations, the \(k\hbox {-}\omega \) shear stress transport (SST) turbulence model, coupled with a two-equation \(\gamma {-Re}_\theta \) transition model, is used for the turbulence closure. Results are presented for a NACA66 hydrofoil undergoing slowly and rapidly pitching motions from \(0^{\circ }\) to \(15^{\circ }\) then back to \(0^{\circ }\) at a moderate Reynolds number \(Re=7.5\times 10^{5}\). The results reveal that the transient flow structures can be observed by the LCS method. For the slowly pitching case, it consists of five stages: quasi-steady and laminar, transition from laminar to turbulent, vortex development, large-scale vortex shedding, and reverting to laminar. The observation of LCS and Lagrangian particle tracers elucidates that the trailing edge vortex is nearly attached and stable during the vortex development stage and the interaction between the leading and trailing edge vortex caused by the adverse pressure gradient forces the vortexes to shed downstream during the large-scale vortex shedding stage, which corresponds to obvious fluctuations of the hydrodynamic response. For the rapidly pitching case, the inflection is hardly to be observed and the stall is delayed. The vortex formation, interaction, and shedding occurred once instead of being repeated three times, which is responsible for just one fluctuation in the hydrodynamic characteristics. The numerical results also show that the FTLE field has the potential to identify the transient flows, and the LCS can represent the divergence extent of infinite neighboring particles and capture the interface of the vortex region.

Graphical Abstract

In this paper, the transient flow structures around a pitching hydrofoil are studied with the FTLE and the LCS. The observation of LCS and Lagrangian particle tracers elucidates the vortex development and interactions. The numerical results also show that the FTLE field has the potential to identify the transient flows, and the ridges of FTLE, LCS, can represent the divergence extent of infinite neighboring particles and capture the interface of the vortex region.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

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
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

References

  1. Ducoin, A., Young, Y.L.: Hydroelastic response and stability of a hydrofoil in viscous flow. J. Fluids Struct. 38, 40–57 (2013)

    Article  Google Scholar 

  2. Kinsey, T., Dumas, G.: Three-dimensional effects on an oscillating-foil hydrokinetic turbine. J. Fluids Eng. 134, 071105 (2012)

    Article  Google Scholar 

  3. Huang, B., Ducoin, A., Young, Y.L.: Evaluation of cavitation models for prediction of transient cavitating flows around a stationary and a pitching hydrofoil. The 8th International Symposium on Cavitation, Singapore (2012)

  4. Shyy, W., Lian, Y., Tang, J., et al.: Computational aerodynamics of low Reynolds number plunging, pitching and flexible wings for MAV applications. Acta Mech. Sin. 24, 351–373 (2008)

    Article  MATH  Google Scholar 

  5. Sheng, W., Galbraith, R.A.M., Coton, F.N.: Prediciton of dynamic stall onset for oscillatory low-speed airfoil. J. Fluids Eng. 130, 101204 (2008)

    Article  Google Scholar 

  6. Fujisawa, N., Shibuya, S.: Observations of dynamic stall on Darrieus wind turbine blades. J. Wind Eng. Ind. Aerodyn. 89, 201–214 (2001)

    Article  Google Scholar 

  7. Yen, J., Ahmed, N.A.: Parametric study of dynamic stall flow field with synthetic jet actuation. J. Fluids Eng. 134, 071106 (2012)

    Article  Google Scholar 

  8. Raffel, M., Kompenhans, J., Wernert, P.: Investigation of the unsteady flow velocity field above an airfoil pitching under deep dynamic stall conditions. Exp. Fluids 19, 103–111 (1995)

    Article  Google Scholar 

  9. McCroskey, W.J.: Unsteady airfoils. Annu. Rev. Fluid Mech. 14, 285–311 (1982)

    Article  Google Scholar 

  10. Carr, L.W.: Progress in analysis and prediction of dynamic stall. J. Aircr. 25, 6–17 (1988)

    Article  Google Scholar 

  11. Ekaterinaris, J.A., Platzer, M.F.: Computational prediction of airfoil dynamic stall. Prog. Aerosp. Sci. 33, 759–846 (1997)

    Article  Google Scholar 

  12. Anderson, J.M., Streitlien, K., Barrett, D.S., et al.: Oscillating foils and high propulsive efficiency. J. Fluids Mech. 360, 41–72 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ducoin, A.: Etude experimentale et numerique du chargement hydrodynamique des corps portants en regime transitoire avec prise en compte du couplage fluide structure. [Ph. D. Thesis], Institut de Recherche de l’Ecole Navale, Lanvéoc Poulmic, Ecole Centrale de Nantes, France (2008) (in French)

  14. Ducoin, A., Astolfi, J.A., Deniset, F., et al.: Computational and experimental investigation of flow over a transient pitching hydrofoil. Eur. J. Mech./B Fluids 28, 728–743 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Arndt, R.E.A.: Cavitation in vortical flows. Annu. Rev. Fluid Mech. 34, 143–175 (2002)

    Article  MathSciNet  Google Scholar 

  16. Foeth, E.J., van Terwisga, T., van Doorne, C.: On the collapse structure of an attached cavity on a three-dimensional hydrofoil. J. Fluids Eng. 130, 071303 (2008)

  17. Gopalan, S., Katz, J.: Flow structure and modeling issues in the closure region of attached cavitation. Phys. Fluids 12, 895–911 (2000)

  18. Ji, B., Luo, X.W., Arndt, R.E.A., et al.: Numerical simulation of three dimensional cavitation shedding dynamics with special emphasis on cavitation–vortex interaction. Ocean Eng. 87, 64–77 (2014)

    Article  Google Scholar 

  19. Ji, B., Luo, X.W., Arndt, R.E.A., et al.: Large eddy simulation and and theoretical investigation of the transient cavitating vortical flow structure around a NACA66 hydrofoil. Int. J. Multiph. Flow 68, 121–134 (2015)

    Article  MathSciNet  Google Scholar 

  20. Menter, F.R.: Improved two-equation \(k\text{- }\omega \) turbulence models for aerodynamic flows. NASA Tech. Memo. 34, 103975 (1992)

  21. Langtry, R.B., Menter, F.R., Likki, S.R., et al.: A correlation-based transition model using local variables-part I: Model formulation. J. Turbomach. 128, 413–422 (2006)

    Article  Google Scholar 

  22. Menter, F.R., Langtry, R.B., Likki, S.R., et al.: A correlation-based transition model using local variables-part II: Test cases and industrial applications. J. Turbomach. 128, 423–434 (2006)

    Article  Google Scholar 

  23. Menter, F.R., Langtry, R.B., Völker, S.: Transition modeling for general purpose CFD codes. Flow Turbul. Combust 77, 277–303 (2006)

    Article  MATH  Google Scholar 

  24. Wu, Q., Wang, G.Y., Huang, B.: Numerical methods and transition investigation of transient flows around a pitching hydrofoil. 6th International Conference on Pumps and Fans with Compressors and Wind Turbines, IOP Conference Series: Materials Science and Engineering 52, 022001 (2013)

  25. Hunt, J.C.R., Wray, A.A., Moin, P.: Eddies, stream, and convergence zones in turbulent flows. Center for Turbulence Research Report CTR-S88, 193–208 (1988)

  26. Haller, G., Yuan, G.: Lagrangian coherent structures and mixing in two-dimensional turbulence. Phys. D 147, 352–370 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rom-Kedar, V., Leonard, A., Wiggins, S.: An analytical study of transport, mixing, and chaos in an unsteady vortical flow. J. Fluids Mech. 214, 347–358 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shadden, S.C., Lekien, F., Marsden, J.E.: Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Phys. D 212, 271–304 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. O’Farrell, C., Dabiri, J.O.: A Lagrangian approach to identifying vortex pinch-off. Chaos 20, 017513 (2010)

    Article  Google Scholar 

  30. Peng, J., Dabiri, J.O.: Transport of inertial particles by Lagrangian coherent structures: application to predator-prey interaction in jelly fish feeding. J. Fluid Mech. 623, 75–84 (2009)

    Article  MATH  Google Scholar 

  31. Tang, J.N., Tseng, C.C.: Lagrangian-based investigation of multiphase flows by finite-time Lyapunov exponents. Acta Mech. Sin. 28, 612–624 (2012)

    Article  MathSciNet  Google Scholar 

  32. Huang, B., Wu, Q., Wang, G.Y.: Numerical simulation of unsteady cavitating flows around a transient pitching hydrofoil. Sci. China Tech. Sci. 57, 101–116 (2014)

    Article  Google Scholar 

  33. Huang, B., Ducoin, A., Yong, Y.L.: Physical and numerical investigation of cavitating flows around a pitching hydrofoil. Phys. Fluids 25, 102109 (2013)

    Article  Google Scholar 

  34. Wang, G., Ostoja-Starzewski, M.: Large eddy simulation of a sheet/cloud cavitation on a NACA0015 hydrofoil. Math. Model. 31, 417–447 (2006)

    Article  Google Scholar 

  35. Adrian, R.J., Meinhart, C.D., Tomkins, C.D.: Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. Seo, J.H., Moon, Y.J., Shin, B.R.: Prediction of cavitating flow noise by direct numerical simulation. J. Comput. Phys. 227, 6511 (2008)

    Article  MATH  Google Scholar 

  37. Lipinski, D., Mohseni, K.: Flow structures and fluid transport for the hydromedusae \(Sarsia\,tubulosa\) and \(Aequorea\,victoria\). J. Exp. Biol. 212, 2436–2447 (2009)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to express our sincere gratitude to Prof. Chien-Chou Tseng (National Sun Yet-sen University), Prof. Young (Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan, USA) and Prof. Antoine Ducoin (French Naval Academy Research Institute (IRENav), France; LHEEA Laboratory, Ecole Centrale de Nantes, Nantes, France) for their helpful comments and support of our work. Completion of this research would not have been possible without their support and understanding. The project was supported by the National Natural Science Foundation of China (Grants 51306020, 11172040), the Natural Science Foundation of Beijing (Grant 3144034) and the Excellent Young Scholars Research Fund of Beijing Institute of Technology.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Biao Huang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, Q., Huang, B. & Wang, G. Lagrangian-based investigation of the transient flow structures around a pitching hydrofoil. Acta Mech. Sin. 32, 64–74 (2016). https://doi.org/10.1007/s10409-015-0484-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-015-0484-8

Keywords

Navigation