同心旋轉圓柱間調制Couette流及Taylor渦旋流之不穩定

Abstract

同心圓柱間的旋轉流場在流體動力學之研究上處於極為重要之一環，其有趣且複雜之現象，至今仍是眾多學者及研究人員相當關注的研究課題。本論文主要目的為利用數值方法建構及分析不同半徑比之同心圓柱間旋轉流場的型態，對於同心圓柱間多變之流況，以雷諾數做為描繪流況之參數，探討此旋轉流場在調制與非調制轉速下之流動特性。 本研究主題包括(1)不同調制效應的Couette流轉換為Taylor渦旋流(2)調制與非調制Taylor渦旋流流場分析(3)非調制Taylor渦旋流轉換為波動Taylor渦旋流。首先，針對在不同的調制振幅及頻率下的Couette流進行穩定性分析研究，發生此不穩定將形成調制Taylor渦旋流。根據Floquet理論可將擾動量分成時間與空間兩部份，其中時間函數與調制具相同的頻率。數值方法採用Galerkin和Collocation法將擾動方程式轉換為代數的特徵方程式，最後以QZ法求解特徵值，此複數型態特徵值的實數部將為流場穩定與否的判斷指標。其次，探討調制與非調制Taylor渦旋流場，在運算上直接求解二維、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，其中數值方法是以Adam-Bashforth法和Crank-Nicolson法分別處理方程式中的非線性及線性項，再將擾動方程式轉換為矩陣方程式並進行求解原始的壓力及速度分量。最後，針對超臨界、軸對稱的非調制Taylor渦旋流進行穩定性分析，求取在不同波數與半徑比下Taylor渦旋流轉換為波動渦旋流的最低穩定曲線，在此階段主要以線性理論來簡化三維擾動的Navier-Stokes方程式以及連續方程式，再利用Galerkin和Collocation等數值方法，將擾動方程式轉換為代數的特徵方程式，如同第一階段Couette流場分析準則，根據複數型態特徵值的實數部作為判斷Taylor渦旋流發生不穩定之依據。

Fluid flow between two concentric rotating cylinders has remained an important topic in fluid dynamics, attracting scholars and researchers to date. In this study, numerical methods are used to analyze and simulate flows and stabilities, which are characterized by a Reynolds number, under different radius ratios and modulated effects. This study focuses on (1) the transition of Couette flow to Taylor vortex flow under different modulated amplitudes and frequencies, (2) the non-modulated and modulated Taylor vortex flows, and (3) the transition of the non-modulated Taylor vortex flow to wavy vortex flow. First, the instability of modulated Couette flow before it transitions to modulated Taylor vortex flow, as well as the effects of modulated amplitude and frequency are studied. By using the Floquet theorem, perturbations are expanded into two series with time periodical coefficients which has the same period as that of the modulation. By following the work of Galerkin and using collocation methods, the equation is transformed into an algebraic eigenvalue problem. The QZ algorithm is employed to solve for the eigenvalues that determine the flow stability. Second, the primitive variables of modulated and non-modulated Taylor vortex flows are solved numerically using the Crank-Nicolson and Adam-Bashforth methods to discretize the linear and nonlinear terms, respectively, of the Navier-Stokes equations. Finally, the stabilities of supercritical Taylor vortex flows are studied by perturbing the nonlinear Taylor vortex flows. Using the same techniques as in part I, the marginal curves of transition to wavy vortex flows are obtained for different radius ratios and axial wave numbers. The resulting stability boundary curve for transition of supercritical Taylor vortex flow is different from that obtained in previous studies, in which the Reynolds number of the inner cylinder is assumed to increase quasi-statically.