以精確色散之Petrov-Galerkin有限元模型求解二維不可壓縮Navier-Stokes方程於移動網格

Abstract

This thesis aims to develop a two-dimensional Petrov-Galerkin (PG) finite element model for effectively resolving erroneous oscillations in the simulation of incompressible viscous fluid flows at high Reynolds numbers in moving meshes by preserving the dispersion relation property. In order to stress the effectiveness of the developed test functions in providing better dispersive nature, we have conducted fundamental studies on the convection-diffusion equation. Several benchmark problems amenable to exact solutions are investigated for the sake of validation. The Navier-Stokes fluid flows in a lid-driven cavity and backward-facing step are also studied at different Reynolds numbers. For the incompressible flow problem with a moving boundary, the Arbitrary Lagrangian-Eulerian (ALE) method is developed in the formulation that is applicable to the time-varying domains. The flow over an oscillating square cylinder is chosen for validation. In order to apply our method in biomechanics area, we consider a vocal fold vibration problem and simplify the geometry as a contraction-and-expansion channel. By virtue of the fundamental analyses and numerical validations, the proposed dispersion-relation-preserving Petrov-Galerkin (DRP-PG) finite element model has been proven to be highly reliable and applicable to solve a wide range of incompressible flow problems in moving meshes.