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.　©　2013　IEEE.