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

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-ω shear stress transport (SST) turbulence model, coupled with a two-equationγ- Reθ transition model, is used for the turbulence closure.Results are presented for a NACA66 hydrofoil undergoing slowly and rapidly pitching motions from 0~?to 15~?then back to 0~?at a moderate Reynolds number Re = 7.5 × 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 observedand 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.     

Analysis of nonlinear oscillations by wavelet transform: Lyapunov exponents

In this paper we give the definition of exponents which would look like Lyapunov exponents in the cases of non-smooth flows of differential equations or iterated maps, and carry back Lyapunov exponents in smooth cases. Here we test our definition by using some simple linear and nonlinear smooth examples.