臺灣博碩士論文加值系統

本論文是利用三維線性穩定性分析，建立一套兩同心圓環柱體且在側向施加水平溫度梯度的系統，設定其內壁固定為高溫，外壁固定為低溫，利用溫度差異所產生的熱對流效應，來觀察其圓環間隙中之對流穩定性。在以往自然對流的討論中，前人對於熱對流的實驗與數值模擬研究許多，不過主要是以軸對稱穩定性分析進行探討，鮮少對非軸對稱且於圓柱座標系中進行討論。而本文透過MATLAB數值分析及穩定性理論，收集大量數據並整理與分析其一系列的關聯性討論。
本研究是運用時間之線性穩定性理論，探討軸對稱與非軸對稱的模式下，觀察兩圓環間隙距離(內外圓柱半徑比，η)、溫度差、普朗特數(Pr)對此系統的不穩定性影響。主要探討軸對稱(l=0)和非軸對稱(l=1~4)在Pr=1～100與η=0.01～0.99間的穩定性。將中性曲線圖之極值數據繪製成穩定性邊界圖，觀察其Grc、kc與Pr、η之穩定性變數，並了解其物理意義。
同時，觀察到中性曲線圖有雙峰的產生，分別是浮力模態(Buoyant mode)與剪力模態(shear mode)，且在不同的圓環間隙距離與普朗特數(Pr)之下，會使得浮力模態(Buoyant mode)與剪力模態(shear mode)相互競爭，而產生雙峰不穩定性。在中性曲線圖中，可以發現臨界波數(kc)之劇烈跳動，這個跳動是因為系統中產生模態的轉變。根據研究結果顯示，非軸對稱模式比軸對稱模式更加穩定，除了在圓環間隙距離很大時，軸對稱模式比非軸對稱(l=1)模式更有優勢，並觀察方位角波長或方位角波數對於此系統的不穩定性影響，且在非軸對稱模式(l=1~4)之穩定性邊界圖中各自都觀察到shear-mode band(SMB)發生。

In this paper, a system of two concentric circular cylinders with horizontal temperature gradients applied laterally is established by using three-dimensional linear stability analysis. The convective stability in the circular gap is observed by setting the inner wall to be fixed at high temperature and the outer wall to be fixed at low temperature and using the heat convection effect generated by the temperature difference. In the previous discussions on natural convection, there are many experimental and numerical simulations on thermal convection, mainly based on the axial symmetric stability analysis, but rarely discussed on the non-axial symmetric and cylindrical coordinate system. In this paper, we collect a large amount of data through MATLAB numerical analysis and stability theory, and organize a series of correlation discussions.
This research use the linear stability theory of time to discuss the axisymmetric and non-axisymmetric modes, observe the gap distance between the two rings,temperature difference, and Prandtl number (Pr) to the system instability effects. The stability of the system is mainly investigated for axial symmetry (l=0) and non-axial symmetry (l=1~4) between Pr=1～100 and η=0.01～0.99. The stability variables of Grc, kc and Pr, η are observed and their physical significance is understood.
At the same time,it is observed that the neutral curves have bimodal peaks, which are buoyant mode and shear mode, and under different circular gap distance and Prandtl number (Pr),the shear mode and buoyant mode will compete with each other and generate bimodal instability.The buoyant mode and shear mode are competing with each other to produce bimodal instability.In the neutral curves,a sharp jump in the critical wavenumber (kc) is observed, and this jump is due to the change of the mode in the system.According to the results,the non-axisymmetric mode is more stable than the axial symmetric mode, except when the circular gap distance is large,the axial symmetric mode is more advantageous than the non-axisymmetric mode, and the effect of azimuthal wavelength or azimuthal wavenumber on the instability of this system is observed, and the occurrence of SMB is observed in the stability boundary diagram for each of the non-axisymmetric modes (l=1~4). The SMB is observed in the stability boundary diagram of the non-axisymmetric mode (l=1~4).

致謝 I
摘要 II
Abstract III
目錄 V
圖目錄 VII
表目錄 VIII
符號說明 IX
第一章 緒論 1
1.1 研究背景 1
1.2 文獻回顧 3
1.3 研究動機 7
1.4 研究方法 8
第二章 理論模型 9
2.1 物理模型 9
2.2 邊界與初始條件、基本假設 10
2.3 Boussinesq approximation 11
2.4 統御方程式 11
第三章 線性穩定性分析 13
3.1 統御方程式之無因次化 13
3.2 統御方程式之求解基態解 15
3.3 微小擾動方程式(small perturbation equation) 19
3.4 正規模態展開(Normal modes expansion) 20
第四章 數值分析 23
4.1 頻譜分析法(Spectral method) 23
4.2 切比雪夫配置法(Chebyshev Collocation method) 24
第五章 結果與討論 27
5.1 程式碼之收斂情形與驗證正確性 27
5.1.1 程式碼之收斂情形 27
5.1.2 程式碼之驗證正確性 27
5.2 參數設定 30
5.3 中性曲線圖與穩定性邊界圖 30
5.3.1 非軸對稱之流動特徵 33
5.3.2 軸對稱與非軸對稱中之shear-mode band(SMB) 34
5.4 普朗特數對於軸對稱與非軸對稱狀態的穩定性影響 34
5.5 軸對稱之穩定性討論 38
5.6 非軸對稱之穩定性討論 42
5.6.1非軸對稱之臨界波數(kc)跳動點 45
第六章 結論與未來展望 50
6.1 結論 50
6.2 未來展望 50
參考文獻 51

1. J. Boussinesq,1903：Théorie Analytique de la Chaleur, Vol. II, Gauthier-Villars, Paris.
2. L. Rayleigh,1916：On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side, Philos. Mag. ,32,p.529-546.
3. H. Jeffreys,1930：The Instability of a Compressible Fluid heated below, Proc. Camb. Phil. Soc.,26,p.170-172.
4. E. A. Spiegel and G. Veronis,1960：On the Boussinesq Approximation for a Compressible Fluid. Astrophysical Journal, 131, p.442
5. J. W. Elder,1965：Laminar free convection in a vertical slot, J. Fluid Mech. , 23,part 1,p.77-98
6. C. M. Vest and V. S. Arpaci,1969：Stability of natural convection in a vertical slot,J. Fluid Mech. 36, 1-15.
7. G. De Vahl., & R. W. Thomas,1969：Natural Convection between Concentric Vertical Cylinders,The Physics of Fluids , 12, II-198
8. J. E. Hart,1971：Stability of the flow in a differentially heated inclined box, J. Fluid Mech. ,47, p.547-576
9. S. A. Korpela & D. Gozum and C. B. Baxis, 1973: On the stability of the conduction regime of natural convection in a vertical slot, Int. J. Heat Mass Transfer 16, p.1683-1690.
10. I. G. Choi and S. A. Korpela,1980：Stability of the conduction regime of natural convection in a tall vertical annulus, J. Fluid Mech. , 99, p.725-738
11. G. B. McFadden, S. R. Coriell, R. F. Boisvert, and M. E. Glicksman, 1984：Asymmetric instabilities in buoyancy-driven flow in a tall vertical annulus, The Physics of Fluids 27, p.1359-1361.
12. Y. M. Chen and A. J. Pearlstein, 1989：Stability of free-convection flows of variable-viscosity fluids in vertical and inclined slots, J Fluid Mech. ,198, p.513-541.
13. B. Brenier, B. Roux and P. Bontoux, 1986：Comparaison des methodes Tau-Chebyshev et Galerkin dans l’etude de stabilite ́ des mouvements de convection naturelle. Proble ̀me des valeurs propres parasites, J. Me ́c. The ́or. ,5, p.95-119.
14. G. D. McBain and S. W. Armfield,2004：Natural convection in a vertical slot: accurate solution of the linear stability equations, ANZIAM J. 45, p.C92-C105.
15. M. Prud’homme and P. L. Que ́re ́, 2007：Stability of stratified natural convection in a tall vertical annular cavity, The Physics of Fluids 19, 094106.
16. T. Seelig, A. Meyer, P. Gerstner, M. Meier, M. Jongmanns, M. Baumann, V. Heuveline, and C. Egbers, 2019：Dielectrophoretic force-driven convection in annular geometry under Earth’s gravity, Int. J. Heat Mass Transfer,139, p.386-398.
17. C. C. Wang and F. Chen, 2022：The bimodal instability of thermal convection in a tall vertical annulus, The Physics of Fluids,34,104102.
18. I. T. Dolapci, 2004：Chebyshev collocation method for solving linear differential equations, Math. Comput. Appl., 9, p.107-115.
19. C. B. Moler, G. W. Stewart, 1973：An algorithm for generalized matrix eigenvalue problems, Society for Industry and Applied Mathematics(SIAM),10, p.241-256.
20. R. F. Bergholz, 1978：Instability of steady natural convection in a vertical fluid layer, J. Fluid Mech. ,84, p.743-768.
21. W. Y. Hunang and F. Chen, 2023：Stability of the double-diffusive convection generated through the interaction of horizontal temperature and concentration gradients in the vertical slot, AIP Conf. Proc.,13,055215
22. C. C. Wang, F. Chen, 2022：On the double-diffusive layer formation in the vertical annulus driven by radial thermal and salinity gradients, Mech. Res. Commun., 125, 103991.