Abstract:
This thesis mainly focuses on computational techniques applied to stabilize unstable Navier-Stokes models. The models arising from the Navier stokes equation are an essential aspect in engineering applications and applied mathematics in fluid mechanics, which is significantly depends on Reynolds number (Re), and if Re≥300, the corresponding model will be unstable. The computation steps are designed to approximate the full models with the ROMs, find the reduced-order feedback matrices, and attain the optimal feedback matrices for stabilizing the desired Navier-Stokes models. The prime concern is exploring the Riccati-based boundary feedback stabilization of incompressible Navier-Stokes flow via Krylov subspace techniques. Since the volume of data derived from the original models is large, the feedback stabilization process through the Riccati equation is always infeasible. Therefore, a H_2 optimal model-order reduction scheme for reduced-order modeling, preserving the sparsity of the system, is required. Some conventional methods exist, but they have some adversities, such as the requirement of high computation time and memory allocation, complex matrix algebra, and uncertainty of the stability of the reduced-order models. To overcome these drawbacks, an extended form of Krylov subspace-based Two-Sided Iterative Algorithm (TSIA) is implemented to stabilize non-symmetric index-2 descriptor systems explored from unstable Navier-Stokes models. The proposed techniques are sparsity-preserving and utilize the Wilson condition to efficiently satisfy the reduced-order modeling approach through the sparse-dense Sylvester equations. To solve the desired Sylvester equations, sparsity-preserving Krylov subspaces are structured via the system of linear equations with a compact form of matrix-vector operations. Inverse projections approaches are applied to get the optimal feedback matrix from reduced-order models. To validate the efficiency of the proposed techniques, transient behaviors of the target systems are observed, incorporating the tabular and figurative comparisons with MATLAB simulations. Finally, to reveal the advancement of the proposed techniques, we compare our work with some existing results. From the tabular and graphical comparisons of the results of numerical computations, it is observed that RKSM is not applicable for the target models due to the non-symmetric structure. In contrast, TSIA can be suitably applied to solve Sparse-dense Sylvester equations for reduced-order modeling. Furthermore, by the TSIA, full models can be efficiently approximated by the corresponding ROMs with minimized H_2 error norm, and the inverse projection scheme is effective in computing the optimal feedback matrices from the reduced-order feedback matrices to stabilize the target models more efficiently than existing methods. Thus, it can be concluded that by utilizing TSIA, unstable Navier-Stokes models can be stabilized with better accuracy and less computing time.
