Reduced Order Modeling And Data Assimilation For The Navier-Stokes Equations

The proper orthogonal decomposition(POD) using Galerkin projection method has been used extensively for reduced order models. The POD method optimally extracts the few most energetic modes from the numerical solutions that can accurately repre-sent the system dynamics. However, the POD/Galerkin finite element model (FEM) lacks stability and spurious oscillations. Another difficulty that arises in applying the POD/Galerkin method to nonlinear fluid problems involves the efficient computation of the projection of the nonlinear terms that are present in the equations. The Four-dimensional variational data assimilation (4D-Var) has been used widely in oceano-graphic and atmospheric models over the last years. However, large dimensionality of the discrete realistic model is the main difficulty in the implementation of4D-Var data assimilation. This thesis is primarily concerned with Stabilisation methods of Re-duced Order Modeling of the Navier-Stokes Equations, Non-linear model reduction of the Navier-Stokes Equations and4D-var based on newly presented POD formulation in this thesis.In this thesis, a new nonlinear Petrov-Galerkin approach has been developed for Proper Orthogonal Decomposition (POD) Reduced Order Modelling (ROM) of the Navier-Stokes equations. The new method is based on the use of the cosine rule between the advection direction in Cartesian space-time and the direction of the gradient of the solution. A finite element pair, P1DGP2, which has good balance preserving properties is used here, consisting of a mix of discontinuous (for velocity components) and continuous (for pressure) basis functions. The results of numerical tests are presented for a wind driven2D gyre and the flow past a cylinder, which are simulated using the unstructured mesh finite clement CFD model in order to illustrate the numerical performance of the method.A new reduced order model based upon proper orthogonal decomposition (POD) for solving the Navier-Stokes equations is presented as well. The novelty of the method lies in its treatment of the equation’s non-linear operator, for which a new method is proposed that provides accurate simulations within an efficient framework. The method itself is a hybrid of two approaches that have already been developed to treat non-linear operators within reduced order models. The first of these approaches is one that approximates non-linear operators through quadratic expansions, and this is then blended within a second technique known as the Discrete Empirical Interpolation Method (DEIM). The method proposed applies the quadratic expansion to provide a first approximation of the non-linear operator, and DEIM is then used as a corrector to improve its representation. In addition to the treatment of the non-linear operator the POD model is stabilized using a Petrov-Galerkin method. This adds artificial dissipation to the solution of the reduced order model which is necessary to avoid spurious oscillations and unstable solutions. A demonstration of the capabilities of this new approach is provided by simulating a flow past a cylinder and gyre problems. Comparisons are made with other treatments of non-linear operators, and these show the new method to provide significant improvements in the solution’s accuracy. A novel POD inverse model, developed for an unstructured adaptive mesh finite element ocean model is presented in this thesis. The reduced adjoint model is derived directly from the discretized reduced petrov-galerkin projection and POD/DEIM based forward model.The computational cost for the four-dimensional variational data assimilation is significantly reduced.