| Summary: | 博士 === 國立臺灣大學 === 工程科學與海洋工程學系 === 91 === The main purpose of this dissertation is to develop finite element models to solve the incompressible Navier-Stokes equations and their related constitutive equations. We classified the governing equations into three types of the prototype equations, named ''convection-reaction equation",
''convection-diffusion equation" and ''convection-diffusion-reaction equation".
By developing finite element models for the prototype equations,
the differential governing equations can be solved.
Within the streamline upwind Petrov Galerkin finite element frame, the convection-diffusion and convection-reaction models were developed. A feature common to the models is that both of them provide nodally exact solutions to the convection-diffusion and the convection-reaction equations in one-dimension. The convection-diffusion model was applied to solve the Navier-Stokes equations. In this model, the bi-quadratic basis function for velocity and bi-linear basis function for pressure were selected in order to satisfy the LBB stability condition. The convection-reaction model was developed to solve the particle trajectory equation in two-phase flow and the constitutive equations for extra stresses in non-Newtonian fluid flow.
By combining a high-order conditionally monotone model and a low order unconditionally monotone model, we developed the composite unconditionally monotone models for the convection-diffusion and the convection-diffusion-reaction equations. By applying the monotone convection-diffusion model to solve the heat and the species concentration equations, the positive valued temperature and the species concentration can be obtained. By applying the monotone convection-diffusion-reaction model to solve the constitutive equations of the turbulent quantities, k and epsilon, in turbulent flow, the positive turbulent quantities including turbulent eddy viscosity, turbulent kinetic energy and turbulent energy dissipation rate can be obtained.
After numerical validations, the proposed models were applied to simulate various flows including the rotating impeller flow, the inhaled particle motion in the human central airway,
non-Newtonian fluid flow, the natural convection problem, the double diffusive problem and the turbulent flow.
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