Nonlinear Galerkin method for low-dimensional modeling of fluid dynamic system using POD modes

Truncation of higher POD modes may describe the behaviors of the system inaccurately. In this context, nonlinear Galerkin method considering effects of higher modes via approximate inertial manifolds is presented to approximate the nonlinear partial differential equations for fluids. The system with infinite dimension is reduced to lower dimension using POD modes. The space spanned by the modes satisfying the boundary condition of the flow field is split into two subspaces, a finite-dimensional one spanned by lower modes and its complement spanned by higher modes. Furthermore, the interaction between the lower modes and higher modes is considered via approximate inertial manifolds. Then, a numerical example of the flow past NACA0012 airfoil for low dimensional modeling analysis is presented to prove the efficiency of the proposed method. The results show that the presented method can give an accurate description for the dynamic behaviors of the system because of considering the effect of higher modes and preserve the topological structure of the system with less computational time, compared with traditional POD method.