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The Lagrangian approach for the two-dimensional incompressible fluid flows has been studied with the help of dynamical systems techniques: Kolmogorov-Arnold-Moser (KAM) theory, stable, unstable manifold structures, and Lagrangian coherent structures (LCSs). For the time-dependent perturbation problems, we analyze in detail the development of transport barriers that play an important role in the transport process. Firstly, the analytical study of KAM theory is adopted to explain the transport and mixing phenomenon in measured and simulated airfoil flow. Then, the flow topology of unsteady flow behind an airfoil is investigated for the low Reynolds number problem. Simulations are carried out based on a particular Finite-Time Lyapunov Exponent (FTLE) technique for the detection of invariant manifolds of the hyperbolic trajectories. Besides, the Characteristic Base Split (CBS) scheme combined with a dual time stepping technique is utilized to simulate such transient flow problems. Thus, in the course of the current research, the role of the velocity phase plot during vortex formation is explored that is highly periodic and resulted in the formation of a stable pattern of manifolds and invariant tori. Hence, the proposed study encouraged the new picture of the vortex shedding and flow separation process. As a conclusion, our results give a better understanding of invariant tori control transport phenomena that will lead to a new heuristic for unsteady flows.

Analysis of Transport and Mixing Phenomenon to Invariant Manifolds Using LCS and KAM Theory Approach in Unsteady Dynamical Systems