二維和三維自由液面流場與對流熱傳問題之數值研究

Numerical Study of Free Surface Flow and Convective Heat Transfer for 2D and 3D Problems

羅德章(Der-Chang Lo)

指導教授 ： 楊德良

國立臺灣大學/工學院/土木工程學研究所/博士(2004年)

https://doi.org/10.6342/NTU.2004.01038

摘要

本論文之主旨在於應用微分積分法和有限元素法，來求解二維和三維速度-渦度列式與原始變數法的奈維爾-史托克斯方程式。
首先，將奈維爾-史托克斯方程式表示成速度-渦度法的形式，接著利用微分積分法來求解二維和三維自然對流的穴室流場。本模式的最大優點是將流場中的速度場、渦度場及溫度場的未知變數，使用一種偶合的方法來求解單一的矩陣系統，因此能精準的求解出流場中的未知變數。相較其他數值方法本文所採用少釭漁磏I數而能得到很好的精度，此外在邊界上的渦度值可利用微分積分法精準的計算出。
再者，本文亦採用有限元素法配合投影法的數值方法來求解三維原始變數法的奈維爾-史托克斯方程式。此法最主要的概念是使用一種分離的離散式、數值平衡擴散式、時間前進法及共軛梯度法來求解。三維穴室流和後向階梯流用來驗證模式的可行性與準確性，接著應用本模式來模擬二邊拖曳的混合對流的穴室流場。
最後，本研究提出一種創新的速度-渦度模式來模擬二維和三維自由液面之流場。使用速度-渦度法之模式，不管是計算內之區域或是在自由液面上皆能完全的消除壓力項，經由一些數值實驗的驗證例子顯示此模式具有很好的精確性，此外藉由計算觀點來述說本模式相較於原始變數法的優勢。

關鍵字

自由液面 ； 熱傳 ； 數值研究

並列摘要

The main topics of this dissertation are to apply the differential quadrature (DQ) method and finite element method (FEM) for modeling two- and three-dimensional Navier-Stokes equations in velocity-vorticity form and primitive variables.
First of all, a novel velocity-vorticity formulation is used for 2D and 3D Navier-Stokes equations and its applications in natural convection cavity flow by using DQ method. The main advantages are that the velocity and vorticity as well as temperature fields can be accurately solved by a coupled solution scheme using a single global matrix for all the unknown variables. Vorticity values are automatically determined more accurately by the use of the DQ method using much coarser meshes compared to other numerical schemes.
Secondly, a projection method based on FEM numerical procedure for solving primitive variables form of the Navier-Stokes equations in three dimensions is undertaken. The key idea is that the proposed numerical scheme involves the operator splitting technique, balance tensor diffusivity (BTD), Runge-Kutta time-stepping method, and a conjugate gradient iterative solver. The lid-driven cavity flow and laminar flow over 3D backward-facing step problems are chosen to test the feasibility and accuracy of the method. Further a mixed convection in a two-sided lid-driven differentially heated cubic cavity is investigated numerically.
Finally, the development of an innovative scheme for simulating 2D and 3D free surface flow is made using velocity-vorticity formulation. With the use of the velocity-vorticity formulation, the computations of the pressure fields in the interior regions and at the free surface of the computational domain are completely eliminated. The efficiency of the proposed numerical scheme for free surface flow is proven to be accurate from the numerical examples considered in the present work. Furthermore the advantage of this formulation with respect to primitive variables formulation is addressed from computational point of view.

並列關鍵字

free surface ； heat transfer ； Numerical study

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