Acta Mechanica Sinica

, Volume 32, Issue 1, pp 64–74 | Cite as

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

  • Qin Wu
  • Biao HuangEmail author
  • Guoyu Wang
Research Paper


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.


Transient flow structure Pitching hydrofoil Finite-time Lyapunov exponent Lagrangian coherent structures 



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.


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Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mechanical and Vechicular EngineeringBeijing Institute of TechnologyBeijingChina

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