摘要
同心旋轉圓柱間不可壓縮之黏性流體所產生的運動型態，在整個流體動力學之研究上，可說是處於極為重要之一環。其有趣且複雜之流況，至今仍是眾多學者及研究人員相當關注的研究課題。
本研究之目的在於利用數值分析方法，模擬部分同心旋轉圓柱間的流動型態以及其相關問題。在數值運算上，直接求解三維、時變並以圓柱座標系統表示的Navier-Stokes方程式以及連續方程式。為分析模擬同心旋轉圓柱間流況的轉變情形，我們利用三階精準之上風差分法以及二階精準之中央差分法，配合交錯分佈之均勻網格之數值方法處理。對於同心圓柱間多變之流況，我們以雷諾數做為描繪流況之參數，探討不同內、外柱轉速下之流動特性。
本文利用以上之數值計算方法解答無限長同心旋轉圓柱間之流況，例如，庫頁特流(Couette flow)、泰勒渦流(Taylor vortex flow)、波狀渦流(Wavy vortex flow)等。本文利用其速度分佈、流線及流動型態，簡要的描述這些流動型式。根據非線性理論分析計算結果，發現由庫頁特流至泰勒渦流之轉變過程為典型之超臨界分歧過程。

Abstract
An incompressible viscous fluid motion between two concentric rotating cylinders is one of the important subjects in fluid dynamics. These flows are interesting and complex and they are still attracted many by scholars and researchers up to day.
The purpose of this study is to use a numerical method to analysis and simulate the flow patterns and relevant issues of the flows between two concentric rotating cylinders. In the numerical operation, which directly solves the Navier-Stokes equations and continuity equation that are three-dimensional and time-dependent, we deal with the cylindrical coordinates. To analysis and simulate the transitions in the flows between the concentric rotating cylinders, we use a numerical approach which cooperates with third-order accurate up-wind scheme and second-order accurate central difference scheme in a system of uniformed staggered grids. We examine the flow patterns as a function of the Reynolds number, which characterizes the features of these flows, under the conditions that the rotating speeds of the inner and outer cylinders both varied.
In this study we use the above mentioned numerical method to simulate the flows that between infinite rotating concentric cylinders, for example, the Couette flow, the Taylor vortex flow and wavy vortex flow, etc. We make a concise description for these flow patterns that are representative by means of their velocity distributions, streamlines and the type of flows. According to the non-linear theory, we find that the transition from Couette flow to Taylor vortex flow is indeed a typical supercritical bifurcation.

目錄
中文摘要……………………………………………………………………i
英文摘要……………………………………………………………………ii
誌謝…………………………………………………………………………iii
目錄…………………………………………………………………………iv
符號索引……………………………………………………………………vi
第一章 緒論
1-1 前言………………………………………………………………1
1-2 文獻回顧…………………………………………………………2
1-3 研究目的…………………………………………………………5
1-4 本文內容…………………………………………………………5
第二章 理論基礎
2-1 控制方程式………………………………………………………8
2-2 控制方程式之無因次化…………………………………………9
2-3 流場邊界及初始條件……………………………………………13
第三章 流況之探討
3-1 同心旋轉圓柱間之流況--同向旋轉……………………………15
3-2 同心旋轉圓柱間之流況--反向旋轉……………………………19
第四章 直接數值模擬
4-1 計算流場網格和變數之配置……………………………………25
4-2 運動方程式之離散………………………………………………25
4-3 壓力場之解答……………………………………………………31
4-4 直接數值模擬之邏輯程序………………………………………33
第五章 數值模擬之結果與分析
5-1 庫頁特流…………………………………………………………34
5-1-1無因次時間t = 50 之流況………………………………35
5-1-2無因次時間t = 2000之流況………………… ……………36
5-1-3無因次時間t = 4000之流況…………………… …………37
5-2 泰勒渦流…………………………………………………………38
5-2-1無因次時間t = 100 之流況………………………………38
5-2-2無因次時間t = 2000之流況……………… ………………41
5-2-3無因次時間t = 4000之流況…………… …………………44
5-3 波狀渦流…………………………………………………………47
5-3-1無因次時間t = 2000之流況…………… …………………48
5-3-2無因次時間t = 4000之流況…………… …………………50
5-3-3無因次時間t = 6000之流況……………… ………………53
5-4 庫頁特流至泰勒渦流之轉變型態分析…………………………56
第六章 結論與建議
6-1 結論………………………………………………………………59
6-2 建議………………………………………………………………60
參考文獻……………………………………………………………………62
附圖…………………………………………………………………………65
附表…………………………………………………………………………124

參考文獻
1.A. Goharzadeh, I. Mutabazi, “Experimental characterization of intermittency regimes in the Couette-Taylor system”, Eur. Phys. J. B 19, 157-162 (2001).
2.B. P. Leonard, “A stable and accurate convective modeling procedure based on quadratic upstream interpolation”, Computer Methods in Applied Mechanics and Engineering. 19, pp. 59-98 (1979).
3.C. D. Andereck, S. S. Liu and H. L. Swinney, “Flow regimes in a circular Couette system with independently rotating cylinders”, J.Fluid Mech., vol.164, pp.155-183 (1986).
4.C. B. Liao, S. J. Jane and D. L. Young, “Numerical simulation of three-dimensional Couette-Taylor flows”, Int. J. Numer. Meth. Fluids 29；827-847 (1999).
5.C. W. Hirt, B. D. Nichols, N. C. Romero, A numerical solution algorithm for transient fluid flows, National Technical Information Service, Springfield (1975).
6.C. Van Atta, “Exploratory measurements in spiral turbulence”, J. Fluid Mech., vol.25, part 3, pp.495-512 (1966).
7.E. M. Sparrow, W. D. Munro and V. K. Jonsson, “Instability of the flow between rotating cylinders: the wide-gap problem”, J. Fluid Mech., vol. 20, part 1, pp. 35-46 (1964).
8.E. R. Krueger, R. C. DiPrima, “The stability of a viscous fluid between rotating cylinders whit an axial flow”, J. Fluid Mech., vol. 19, pp.528-538 (1964).
9.E. L. Koschmieder, Benard Cells and Taylor Vortices, Cambridge Univ. Press, (1993).
10.G. Ahlers, D. S. Cannell and M. A. Dominguez Lerma, “Possible mechanism for transitions in wavy Taylor-vortex flow”, Phys. Review A, vol. 27, NO. 2, pp.1225-1227 (1983).
11.Li-Hua Zhang, H. L. Swinney, “Nonpropagating oscillatory modes in Couette-Taylor flow”, Phys. Review A, vol. 31, NO.2, pp. 1006 — 1009 (1985).
12.M. Gorman, H. L. Swinney, “Spatial and temporal characteristics of modulated waves in the circular Couette system”, J. Fluid Mech., vol. 117, pp.123-142 (1982).
13.N. K. Ghaddar, K. Z. Korczak, B. B. Mikic and A. T. Patera, “Numerical investigation of incompressible flow in grooved channel：Part 1. Stability and self-sustained oscillations”, J. Fluid Mech, 163；99-127（1986）.
14.P. R. Fenstermacher, H. L. Swinney and J. P. Gollub, “Dynamical instabilities and the transition to chaotic Taylor vortex flow”, J. Fluid Mech., vol. 94,pp.103-128 (1979).
15.T. T. Lim, Y. T. Chew and Q. Xiao, “A new flow regime in a Taylor-Couette flow”, Phys. Fluids, vol. 10, NO. 12, pp. 3233-3235 (1998).
16.T. T. Lim, Y. T. Chew and Q. Xiao, “Second Taylor vortex flow：Effects of radius ratio and aspect ratio”, Phys. Fluids, vol.14, NO.4, pp.1537-1539 (2002).
17.詹淑菁,“兩旋轉圓筒中三維流場之直接數值模擬”，私立逢甲大學土木系水利工程研究所碩士學位論文，民國八十六年六月。