Date

2022-6-28

Author

DEMİR, Medine

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In this thesis, second-order, efficient and reliable numerical stabilization methods are considered for approximating solutions to the incompressible, viscous fluid flow driven by the Navier-Stokes equations and for the Darcy-Brinkman equations driven by double-diffusive convection. The standard Galerkin finite element method remains insufficient for accurately solving these complex nonlinear equations that creates some problems such as numerical instabilities and unphysical oscillations in the solution. A good numerical algorithm should resolve all the scales in the solution to avoid these problems which requires too much computational effort. Thus, developing proper and efficient numerical algorithm that exhibits correct physical behaviour of the flow and accurately approximates solutions over a finite time interval remains a great challenge in computational fluid dynamics. First, this thesis proposes a numerical scheme which tests and analyzes a subgrid artificial viscosity method to model the incompressible Navier-Stokes equations along a linearly extrapolated BDF2 time discretization method. The method considers the viscous term as a combination of the vorticity and the grad-div stabilization term. The method introduces global stabilization by adding a term, then antidiffuses through the extra mixed variables. A detailed analysis of conservation laws, including both energy and helicity balance of the method is presented. It is shown that the approximate solutions of the method are unconditionally stable and optimally convergent. Several numerical tests are presented for validating the support of the derived theoretical results. Second, this thesis considers the backward Euler based linear time filtering method for the developed energy-momentum-angular momentum conserving formulation of the time dependent, incompressible Navier-Stokes equations in the case of weakly enforced divergence constraint. The method adds time filtering as a post-processing step to the energy-momentum-angular momentum conserving formulation to enhance the accuracy and to improve the approximate solutions. It is shown that in comparison with the backward-Euler based energy-momentum-angular momentum conserving formulation without any filter, the proposed method not only leads to a 2-step, unconditionally stable and second order accurate method but also increases numerical accuracy of solutions. Numerical studies verify the theoretical findings and demonstrate preeminence of the proposed method over the unfiltered case. Third, this thesis studies an efficient, accurate, effective and unconditionally stable time stepping scheme for the Darcy-Brinkman equations in double-diffusive convection. The stabilization within the proposed method uses the idea of stabilizing the curvature for velocity, temperature and concentration equations. Accuracy in time is proven and the convergence results for the fully discrete solution of problem variables is given. Several numerical examples including a convergence study are provided that support the derived theoretical results and demonstrate the efficiency and the accuracy of the method.

Subject Keywords

Subgrid artificial viscosity model, Navier-Stokes equations, linearly extrapolated BDF2, energy-momentum-angular momentum conserving formulation, time filter, Darcy-Brinkman equations, curvature stabilization, finite element method.

URI

https://hdl.handle.net/11511/98154

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Graduate School of Natural and Applied Sciences, Thesis