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The projective integration method based on the Galerkin-free framework with the assistance of proper orthogonal decomposition (POD) is presented in this paper. The present method is applied to simulate two-dimensional incompressible fluid flows past the NACA0012 airfoil problem. The approach consists of using high-accuracy direct numerical simulations over short time intervals, from which POD modes are extracted for approximating the dynamics of the primary variables. The solution is then projected with larger time steps using any standard time integrator, without the need to recompute it from the governing equations. This is called the online projective integration method. The results by the projective integration method are in good agreement with the full scale simulation with less computational needs. We also studythe individual function of each POD mode used in the projective integration method. It is found that the first POD mode can capture basic flow behaviors but the overall dynamic is rather inaccurate. The second and the third POD modes assist the first mode by correcting magnitudes and phases of vorticity fields. However, adding the fifth POD mode in the model leads to some incorrect results in phase-shift forms for both drag and lift coefficients. This suggests the optimal number of POD modes to use in the projective integration method

Online Projective Integral with Proper Orthogonal Decomposition for Incompressible Flows Past NACA0012 Airfoil