許文翰臺灣大學：工程科學及海洋工程學研究所徐銘辰Hsu, Ming-ChenMing-ChenHsu2007-11-262018-06-282007-11-262018-06-282005http://ntur.lib.ntu.edu.tw//handle/246246/51126本研究的目的是發展一保有色散關係之二維Petrov-Galerkin有限元模型，以期有效地去除錯誤的數值振蕩。此外，本文將任意的Lagrangian-Eulerian描述引入Navies-Stokes方程中，以便在移動的網格上得以求解高雷諾數之不可壓縮流體。為了強調所提出之精確色散捕捉能力，我們進行對流–擴散模型方程之基本分析。透過一系列的測試題目，得以證實本文所提出具保有色散關係之Petrov-Galerkin有限元模型確能有效地求解具移動網格問題。最後並將所開發之 DRP-PG有限元模型應用於生醫領域中有關聲帶震動的模擬，以瞭解聲道中流場所引起之發聲機制。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.Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii 1 Introduction 1 1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Working equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Working equations on fixed meshes . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Working equations on moving grids . . . . . . . . . . . . . . . . . . . . . 2 1.3 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.1 Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.2 Petrov-Galerkin (PG) model . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.3 Dispersion-relation-preserving theory . . . . . . . . . . . . . . . . . . . 5 1.3.4 Arbitrary Lagrangian-Eulerian (ALE) method . . . . . . . . . . . . . . . . 5 1.4 Outlines of this study . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Finite element model for two-dimensional convection-diffusion scalar equation . 8 2.1 Transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Interpolation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Weak statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Two-dimensional dispersion-relation-preserving Petrov-Galerkin model . . . . . 13 3.1 Petrov-Galerkin (PG) model . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Dispersion-relation-preserving PG model . . . . . . . . . . . . . . . . . . . 15 3.3 Dispersion and Fourier (or von Neumann stability) analyses . . . . . . . . . 18 3.4 Fundamental studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.5 Numerical studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.5.1 Gaussian and hyperbolic tangent problem . . . . . . . . . . . . . . . . . . 21 3.5.2 Skew convection-diffusion problem . . . . . . . . . . . . . . . . . . . . . 22 3.5.3 Convection-diffusion problem of Smith and Hutton . . . . . . . . . . . . . 23 3.5.4 Rotation of a cone-shaped scalar field . . . . . . . . . . . . . . . . . . 23 3.5.5 Mixing of hot and cold fluids . . . . . . . . . . . . . . . . . . . . . . . 24 4 Two-dimensional incompressible Navier-Stokes equations 48 4.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 Numerical studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2.1 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2.2 Lid-driven cavity flow problem . . . . . . . . . . . . . . . . . . . . . . 52 4.2.3 Backward-facing step flow problem . . . . . . . . . . . . . . . . . . . . . 52 5 Moving boundary problem 68 5.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.2 Geometric conservation law and grid velocity . . . . . . . . . . . . . . . . 69 5.3 Numerical studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.3.1 Flow over a square cylinder . . . . . . . . . . . . . . . . . . . . . . . . 71 5.3.2 Flow over a square cylinder oscillating along the x direction . . . . . . . 73 5.3.3 Contraction-and-expansion channel . . . . . . . . . . . . . . . . . . . . . 75 6 Concluding remarks 96 A DRP-PG coefficients τi 98 B Coefficient matrices for Φx 100 C Derivation for ki and kr 1028732442 bytesapplication/pdfen-US不可壓縮高雷諾數保有色散關係移動網格聲帶two-dimensionalincompressiblehigh Reynolds numbersdispersion-relation- preservingArbitrary Lagrangian-Eulerianvocal foldsthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/51126/1/ntu-94-R92525018-1.pdf