Issues in low-dimensional modeling of unsteady flows: Convergence, asymptotic stability and reconstruction procedures

In the first part of the thesis, the study of long term integration of the low-dimensional system derived from the Navier-Stokes equations is presented. The study is done by employing bifurcation analysis. It is found that the attractors of the low-dimensional system are different from the ones of the direct numerical simulation (DNS). An ad-hoc model has been proposed to correct the long-term dynamics, based on mode-dependent spectral viscosity. The viscosity coefficient is rigorously chosen through bifurcation analysis.;In the second part, a new low-dimensional model developed to accommodate the data with time-dependent boundary conditions is presented. A penalty method has been successfully implemented to deal with such boundary conditions. The sensitivity of the model to the penalty parameter is studied through the stability of the periodic solution. It is found that there is a threshold value of the penalty parameter above which asymptotic stability of the periodic solution is guaranteed. Also, there is a specific range of penalty parameters within which the solution is most accurate.;In the third study, we turn to the application of POD basis in the Galerkin-free model. By employing the POD space as a macroscopic level together with the DNS as a microscopic level, we can employ the projective integration method for the Navier-Stokes equations. An error analysis for this Galerkin-free equation-free POD model is provided and numerical demonstration is performed for both periodic flow and quasi-periodic flow, and verification of the error analysis is also achieved. It is found that this approach can successfully resolve complex flow dynamics at a reduced computational cost and that it can capture the long-term asymptotic state of the flow in cases in which traditional Galerkin-POD models fail.;In the last study, we focused on data assimilation. Some data assimilation methods that utilize the proper orthogonal decomposition in the algorithm together with Kriging interpolation are studied. It is found that with sufficient temporal resolution, the extended-EOF interpolation and extended-Sirovich procedure perform better than the Kriging interpolation and the standard EOF. Also, the computational cost of the extended-EOF interpolation is less expensive than the extended-Sirovich procedure. However, with insufficient temporal resolution or data with black zones, the Kriging interpolation is the most effective method to use.