A numerical method is proposed to approach the Approximate Inertial Man-ifolds(AIMs)in unsteady incompressible Navier-Stokes equations,using multilevel fi-nite element method with hierarchical basis functions.Following AIMS,the unknown variables,velocity and pressure in the governing equations,are divided into two com-ponents,namely low modes and high modes.Then,the couplings between low modes and high modes,which are not accounted by standard Galerkin method,are consid-ered by AIMs,to improve the accuracy of the numerical results.Further,the multilevel finite element method with hierarchical basis functions is introduced to approach low modes and high modes in an efficient way.As an example,the flow around airfoil NACA0012 at different angles of attack has been simulated by the method presented,and the comparisons show that there is a good agreement between the present method and experimental results.In particular,the proposed method takes less computing time than the traditional method.As a conclusion,the present method is efficient in numer-ical analysis of fluid dynamics,especially in computing time.
The lobe dynamics andmass transport between separation bubble andmain flow in flow over airfoil are studied in detail,using Lagrangian coherent structures(LCSs),in order to understand the nature of evolution of the separation bubble.For this problem,the transient flow over NACA0012 airfoil with low Reynolds number is simulated numerically by characteristic based split(CBS)scheme,in combination with dual time stepping.Then,LCSs and lobe dynamics are introduced and developed to investigate themass transport between separation bubble andmain flow,from viewpoint of nonlinear dynamics.The results show that stablemanifolds and unstable manifolds could be tangledwith each other as time evolution,and the lobes are formed periodically to induce mass transport between main flow and separation bubble,with dynamic behaviors.Moreover,the evolution of the separation bubble depends essentially on themass transportwhich is induced by lobes,ensuing energy andmomentum transfers.As the results,it can be drawn that the dynamics of flow separation could be studied using LCSs and lobe dynamics,and could be controlled feasibly if an appropriate control is applied to the upstream boundary layer with high momentum.