研究內容分為數值預測與實驗觀測兩部分。數值預測採有限差分法，探討充滿低溫水之二維與三維高直立容器內之自然對流轉變過程。二維數值模擬分別針對直立同心圓環與直立矩型容器，分析在密度逆轉參數為0.4與0.5下，高寬度比為8，直立同心圓環之半徑比為2，逐步調高Ra值的暫態自然對流轉變現象。研究之Ra值範圍介於 與 之間。並對相同高寬度比的三維直立矩型容器，密度逆轉參數為0.5下，探討其深寬度比的效應以及不穩定機制，研究之Ra值範圍介於 與 。數值結果藉由分析流動結構與溫度分佈、擾動振幅等值線分佈與擾動動能的型態及其分佈，深入探討導致自然對流不穩定的機構。實驗觀測部分設計高直立矩型封閉容器模型之實驗，以液晶顯像法觀察最大密度線的擺動，並以液晶為追蹤物進行流場觀測，實驗結果與數值預測結果相互驗證。
數值結果顯示當低溫水之密度逆轉現象出現，使最大密度線落在容器中間時，引發兩反向環流的產生，兩環流之間的交互作用，源自壁面噴流會合之剪力不平衡（Kelvin-Helmtloz 不穩定），並導致最大密度線扭曲，形成不穩定的密度分佈（類似 Rayleigh/Benard 不穩定），此種不穩定並影響垂直壁之邊界層流動，使邊界層流動不穩定（Tollmien Schlichting 不穩定）提前發生，形成向上波動現象。不論是矩型封閉容器或直立圓環容器，均因其最大密度線位置不同，其不穩定現象隨 Ra 提高亦有所差異，但其不穩定性仍與密度逆轉參數有較大關係。三維模型中，由流動結構、溫度分佈、向上傳遞的波動現象、主頻的關係與擾動動能的貢獻發現，不穩定現象的機制，在二維與三維的結果是一致的。由擾動動能的分析顯示，提供不穩定流動之能量源自浮力。
實驗觀測結果顯示，在發展的過程中，順時環流發展伴隨最大密度位置的移動，以及主環流內的多重環流的成長、合併移動等現象，均可由實驗的流場結構與液晶顯像描述，實驗結果顯示密度逆轉現象對低溫水的自然對流有重大影響。而理論預測與實驗結果比較，相當吻合。

The present investigation consists of numerical prediction and/or experimental visualization on the transition of natural convection of cold water in vertical enclosures. Using a finite difference method, numerical simulations of the transient natural convection of cold water contained in vertical enclosures of height/width ratio 8 (cylindrical annulus or rectangular enclosure) have been performed by increasing the Rayleigh number step by step under two different values of the density inversion parameter of 0.4 and 0.5, respectively. The range of the Rayleigh number considered is from to . The radius ratio of the cylindrical annulus is fixed at 2. The three dimensional numerical simulation focus on the natural convection of cold water in the rectangular enclosure under the density inversion parameter of 0.5 to explore the effect of the depth/width ratio of the enclosure on the instability mechanism. The range of the Rayleigh number considered for the three-dimensional simulations is from to . The evolutions of fluid flow structure and temperature distribution of cold water in the enclosures are visualized numerically by means of contour maps of streamline and/or velocity vector field and isotherm. Furthermore, the instability mechanism responsible for the transition to unsteady natural convection of cold water and its spatial distribution within the enclosures are illustrated by means of the contour maps of fluctuation magnitude of flow variables or compositions of the fluctuating kinetic energy. The experimental work in the present study mainly focuses on the flow and temperature visualizations, by means of a thermochromic liquid crystal, in a test cell, which was designed and constructed to mimic the physical configuration considered in the numerical simulations.
The results of numerical simulation clearly reveal that the value of density inversion parameter predominantly determines the profile of maximum density contour, which demarcates the two contra-rotating circulation flow structures of cold water prevailed in the enclosure. The instability mechanism appears to initiate by the Kelvin-Helmholtz instability due to shear imbalance along the sinking jet flow structure emanating from the ceiling of enclosure, which results in wavy distortion of the maximum density contour of cold water. As a result of the wavy maximum density contour, unstable density stratification (the Rayleigh-Benard instability) is induced. The vertical boundary layer flows along the vertical walls are thereby perturbed and the Tollmien-Schlichting instability arises accordingly. The laminar transition to self-sustained oscillatory convection has a strong bearing with the density inversion phenomenon of cold water. The results of two- or three-dimensional numerical simulations reveal similar mechanism for the unsteadiness of cold water in the rectangular enclosure. The production of fluctuating kinetic energy is completely contributed by the buoyancy.
The results of experimental visualization show that growing, merging and upward-drifting of the secondary vortex is accompanied wavy movement with the maximum density contour. Comparison between the numerical prediction and the experimental visualization reveals a fair agreement.

封面
中文摘要
英文摘要
誌謝
目錄
符號說明
表目錄
圖目錄
第一章 緒論
1-1 研究背景與動機
1-2 文獻回顧
1-3 本文架構
第二章 低溫水在高直立封閉容器內暫態自然對流之轉變--由穩態過渡至週期性流動
2-1 物理模型措述與基本假設
2-2 統御方程式
2-3 數值離散方法
2-4 解題方法與程序
2-5 數值方法正確性之測試
2-6 格點測試
2-7 結果與討論
2-7.1 直立圓環
2-7.2 直立矩型容器
2-7.3 半徑比效應的探討
2-8 結論
第三章 低溫水在高直立矩型封閉容器內暫態自然對流三維數值研究一由穩態過渡至週期性流動
3-1 物理模型與數學模式
3-2 數值方法與測試
3-3 數值方法正確性之測試
3-4 結果與討論
3-4.1 A =2 三維流動結構與溫度分佈
3-4.2 深寬度(A )的效應
3-4.3 平均熱傳係數( )與Ra的關係
3-5 結論
第四章 低溫水在高直矩型容器暫態自然對流之實驗量測與數值預測
4-1 實驗模型與實驗方法
4-2 實驗步驟
4-3 數值模擬
4-4 結果與討論
4-4.1 發展期間流動結構與溫度分佈
4-4.2 穩態或週期流動結構與溫度分佈
4-4.3 熱電偶電度與數值結果的比較
4-5
圖表
其他

Akino, N., Kunugi, T., Ichimiya, K., Mitsushi ro, K., and Ueda, M., "Improved liquid crystal thermometry excluding human color sensation," J. Heat Transfer, Vol. 111, pp. 558-565, 1989.
Batchelor, G. K., "Heat transfer by free convection across a closed cavity between vertical boundies at different temperature, Quart. Appl. Math., Vol. 12, pp. 209-233, 1954.
Bejan, A., "Note on Gill''s solution for free convection in a vertical enclosure," J. Fluid Mech. vol. 90, part 3, pp. 561-568, 1979.
Bejan, A., Convection Heat Transfer, Jone Wiley & Sons, 1984.
Bergholz, R. F., "Instability of steady natural convection in a vertical fluid layer," J. Fluid Mech. vol. 84, part 4, pp. 743-768, 1978.
Bhushan, R., Keyhani, M., Christensen, R. N., and Kulacki, F. A., "Correlation equations for free convection in a vertical annulus with constant heat flux on the inner wall," J. Heat Transfer, vol. 105, pp. 910- 912, 1983.
Braga, S. L. and Viskanta, R., "Transient natural convection of water near its density extremum in a rectangular cavity," Int. J. Heat Mass Transfer, Vol. 35, No. 4, pp. 861-875, 1992.
Braga, S. L. and Viskanta, R., "Effect of density extremum on the solidification of water on the vertical wall of a rectangular cavity, Experimental Thermal and Fluid Science, No. 5, pp. 703-713, 1992.
Briggs, D. G., and Jones, D. N., "Two-Dimensional Periodic Natural Convection in a Rectangular Enclosure of Aspect Ratio One," J. Heat Transfer, vol. 107, pp. 850-854, 1985.
Braun, M. J., Drodzo, M., and Lattime, S. B., "Automatic computer based non-instrusive temperature measurements in laminar natural convection using TLC in enclosure with variable aspect ratio," ASME-FED-Vol 172, Experimental and Numerical Flow Visualization, pp. 111-119, 1993.
Chen, Falin and Wu, C. H., "Unsteady convective flows in a vertical slot containing variable viscosity fluids," Int. J. Heat Mass Transfer, vol. 36, No. 17, pp. 4233-4246, 1993.
Chen, C. F. and Thangam, S., "Convective stability of a variable-viscosity fluid in a vertical slot," J. Fluid Mech., vol. 161, pp. 161-173, 1985.
Chenoweth, D. R. and Paolucci, S., "Natural convection in an enclosed vertical air layer with large horizontal temperature difference," J. Fluid Mech., Vol. 169, pp. 173-210, 1986.
Choi, I. G. and Korpela, S. A., "Stability of the conduction regime of natural convection in tall vertical annulus," J. Fluid Mech., Vol. 99, part 4, pp. 725-738, 1980.
de Vahl Davis, G., and Jones, I. P., "Natural convection in a square cavity : a comparison exercise, Int. J. Num. Meth. Fluids, Vol. 3, pp. 227-248, 1983.
de Vahl Davis, G., and Thomas, R. W., "Natural convection between concentric vertical cylinders, high speed computing in fluid dynamics, Phys. Fluids, Suppl. II, pp. 198-207, 1969.
Elder, J. W., "Laminar free convection in a vertical slot," J. Fluid Mech., vol. 23, part 1, pp. 77-98, 1965a.
Elder, J. W., "Turbulent free convection in a vertical slot," J. Fluid Mech., vol. 23, part I, pp. 99-111, 1965b.
Eckert, E. R. G. and Carlson, W. O., "Natural convection in an air layer enclosed between two vertical plates with different temperatures," Int. J. Heat Mass Transfer, Vol. 2, pp. 106-120, 1961.
Forbes, R. E. and Cooper, J. W., "Natural convection in a horizontal layer of water cooled from above to near freezing," J. Heat Transfer, Vol. 97, pp. 47-53, 1975.
Fusegi, T., Hyun, J.M., Kuwahara, K., and Farouk, B., "A numerical study of three-dimensional natural convection in a differentially heated cubical enclosure, Int. J. Heat Mass Transfer. vol. 34, No. 6, pp. 1543- 1557, 1991.
Fusegi, T., Hyun, J. M., and Kuwahara, K., "Three-dimensional numerical simulation of periodic natural convection in a differentially heated cubical enclosure," Applied Scientific Research, Vol. 49, pp. 271-282, 1992.
Gebhart, B. and Mollendorf, J., "A new density relation for pure and saline water," Deep Sea Research, Vo1.24, pp. 831-838, 1974.
Gill, A. E., "The boundary-layer regime for convection in a rectangular cavity," J. Fluid Mech. vol. 26, part 3, pp. 515-536, 1966.
Gill, A. E., and Davey, A., "Instabilities of a buoyancy-driven system," J. Fluid Mech. vol. 35, part 4, pp. 775-798, 1969.
Gill, A. E., and Kirkham, C. C., "A note on the stability of convection in a vertical slot," J. Fluid Mech. vol. 42, part 1, pp. 125-127, 1970.
Henkes, R. A. W. M. and Hoogendoorn, C. J., "On the stability of the natural convection flow in a square cavity heated from the side," Applied Scientific Research, Vol. 47, pp. 195-220, 1990a.
Henkes, R. A. W. M. and Hoogendoorn, C. J., "Bifuraction to unsteady natural convection for air and water in a cavity heated from side," Heat Transfer Conf., Vol. 2, 1990b.
Heieh, S. S., and Wang, C. Y., "Experimental study of three-dimensional natural convection in enclosures with different working fluids," Int. J. Heat Mass Transfer, Vol. 37, No. 17, pp. 2687-2698, 1994.
Ho, C. J. and Lin, Y. H., "Natural convection of cold water in a vertical annulus with constant heat flux on the inner water," J. Heat Transfer, Vol. 112, pp. 117-123, 1990.
Ho, C. J. and Lin, Y. H., "Laminar natural convection of cold water enclosured in a horizontal annulus with mixed boundary conditions," Int. J. I Heat Mass Transfer, Vol. 31, No. 10, pp. 2113-2121, 1988a.
Ho, C. J. and Lin, Y. H., "Natural convection heat transfer of cold water within an eccentric horizontal cylindrical annulus," J. Heat Transfer, Vol. 110, pp. 894-900, 1988b.
Inaba, H. and Fukada, T., "Natural convection in an inclined square cavity in regions of density inversion of water," J. Fluid Mech., Vol. 142, pp. 363-381,1984a.
Inaba, H. and Fukada, T., "An experimental study of natural convection in an inclined rectangular cavity filled with water at its density extremum," J. Heat Transfer, Vol. 106, pp. 109-115,1984b.
Ivey, G. N., "Experimental on transient in a cavity," J. Fluid Mech., Vol. 144, pp. 389-401, 1984.
Ivey, G. N. and Hamblin, P. F., "Convection near the temperature of maximum density for high Rayleigh number, low aspect ratio, rectangular cavities," J. Heat Transfer, Vol. 111, pp. 100-105, 1989.
Janssen, R. J. A., Henkes, R. A. W. M., and Hoogendoom, C. J., "Transition to time-periodicity of a natural-convection flow in a 3D differentially heated cavity," Int. J. Heat Mass Transfer. vol. 3~, No. II, pp. 2927-2940, 1993.
Janssen, R. J. A., and Henkes, R. A. W. M., "Influence of Prandtl number on instability mechanisms and transition in a differentially heated square cavity," J. Fluid Mech. vol. 290, pp. 319-344, 1995.
Janssen, R. J. A., and Henkes, R. A. W. M., "The first instability mechanism in differentially heated cavities with conducting horizontal walls," J. Heat Transfer, vol. 117, pp. 626-633, 1995.
Janssen, R. J. A., and Henkes, R. A. W. M., "Instabilities in three- dimensional differentially-heated cavities with adiabatic horizontal walls, Phys. Fluids, Vol. 8, No. 1, 1996.
Jeevaraj, C. G. and Patterson, J. C., "Experimental study of transient natural convection of glyceral-water mixtures in a side heated cavity," Int. J. Heat Mass Transfer, Vol. 35, No. 6, pp. 1573-1587, 1992.
Jones, D. N., and Briggs, D. G., "Periodic Two-Dimensional Cavity Flow: Effect of Linear Horizontal Thermal Boundary Condition," J. Heat Transfer, vol. 111, pp. 86-91, Feb., 1989.
Keyhani, M., Kulacki, F. A., and Christensen, R. N., "Free convection in a vertical annulus with constant heat flux on the inner wall," J. Heat Transfer, vol. 105, pp. 454-459, 1983.
Kirkpatrick, A., "Visualization of diffuser outlet flow using liquid crystal sheets," ASHRAE, pp. 159-164, 1995.
Kline, S. J., and McClintock, F. A., "Describing uncertainities in single- sample experiments," Mech. Eng., vol. 75, pp. 3-8, 1953.
Korpela, S. A., Gozum, D., and Baxi C. B., "On the stability of the conduction regime of natural convection in a vertical slot," Int. J. Heat Mass Transfer, vol. 16, pp. 1683-1690, 1973.
Korpela, S. A., Lee, Yee, and Drummond, J. E., "Heat transfer through a double pane window," J. Heat Transfer, vol. 104, pp. 539-544, 1982.
Lankford, K. E., and Bejan, A., "Natural convection in a vertical enclosure filled with water near 40C," J. Heat Transfer, vol. 108, pp. 755- 763, 1986.
Lauriat, G., and Desrayaud, G., "Influences of the boundary conditions and linearization on the stability of a radiating fluid in a vertical layer," Int. J. Heat Mass Transfer. vol. 28, No. 8, pp. 1613-1617, 1985.
Le Quere, P., "A note on multiple and unsteady solutions in two- dimensional convection in a tall cavity," J. Heat transfer, Vol. 112, pp. 965-974, 1990a.
Le Quere, P., "Transition to unsteady natural convection in a tall water- filled cavity," Phys. Fluids (A), Vol. 2, pp. 503-515, 1990b.
Le Quere, P., "Onset of unsteadiness, routes to chaos and simulations of chaotic flows in cavities heated from the side: a review of present status, Heat Transfer Conf., vol. 1, pp. 281-296, 1994.
Le Quere, P. and Alziary de Roquefort, T., "Computation of natural convection in a two-dimension cavity with Chebyshev polynominals," J. Comp. Phys., Vo1.57, pp. 210-228, 1985.
Le Quere, P. and Pecheux, J., "Numerical simulation of multiple flow transitions in a axisymmetric annulus convection," J. Fluid Mech., Vol. 206, pp.517-544, 1989.
Lee, Y,, Korpela, S. A. and Home, R. N., "Structure of multi-cellular natural convection in tall vertical annulus," Proceedings 7th Int. Heat Transfer Conf., Vol. 2, pp. 221-226, Munich, Paper NC 17, 1982.
Lee,-Y. and Korpela, S. A., "Multicellular natural convection in a vertical slot," J. Fluid Mech., Vol. 126, pp. 91-121, 1983.
Leonard, B. P., "A stable and accurate convective modeling procedure based on quadratic upstream interpolation," Comput. Meth. Appl. Mech. Engng., Vol. 19, pp. 59-98, 1979.
Leonard, B. P., "Elliptic systems : Finite difference method IV," in Handbook of Numerical Heat Transfer, W. J. Minkowycz, E. M. Sparrow, G. E. Schneider, and R. H. Fletcher, eds., Wiley, New York, pp. 347-378, 1988.
Lin, D. S. and Nansteel, M. W., "Natural convection heat transfer in a square enclosure containing water near its density maximum," Int. J. Heat Mass Transfer, Vol. 30, No. 11, pp. 2319-2329, 1987a.
Lin, D. S. and Nansteel, M. W., "Natural convection in a vertical annulus containing water near the density maximum," J. Heat Transfer , Vol. 112, pp. 899-905, 1987b.
Mallinson, G. D., and Vahl Davis, G., "Three-dimensional natural convection in a box: a numerical study," J. Fluid Mech., vol. 83, part 1, pp. 1-31, 1977.
McDonough, M. W., and Faghri, A., "Experimental and numerical analyses of the natural convection of water through its density maximum in a rectangular enclosure," Int. J. Heat Mass Transfer. vol. 37, No. 5, pp. 783-801, 1994.
McFadden, G. B., Coriell, S. R., and Boisvert, R. F., "Asymmetric instabilities in buoyancy-driven now in a tall vertical annulus," Phys. Fluids, Vol. 27, No. 6, 1984.
Moffat, R. J., "Contributions to the theory of single-sample uncertainity analysis," J. Fluids Eng., Vol. 104, pp. 250-256, 1982.
Nansteel, M. W., Medjani, K. and Lm, D. S., "Natural convection of water near its density maximum in a rectangular enclosure : low Rayleigh number calculations," Physics Fluids, Vol. 30(2), pp. 312-317,1987.
Nguyen, T. H., Vasseur, P. and Robillard, L., "Natural convection between horizontal concentric cylinder with density inversion: of water for low Rayleigh," Int. J. Heat Mass Transfer, Vol. 97, pp. 1559-1568, 1982.
Nishimura, T., Fujiwara, M., Horie, N., and Miyashita, H., "Temperature visualizations by use of liquid crystals of unsteady natural convection during supercooling and freezing of water in an enclosure with lateral cooling," Int. J. Heat Mass Transfer. vol. 34, No. 10, pp. 2663-2668, 1991.
Nishimura, T., Fujiwara, M., and Miyashita, H., "Visualization of temperature fields of transient natural convection with maximum density effect in a water-filed enclosure by chiral nematic liquid crystals," J. Chem. Eng. Japan, Vol. 23, No. 2, pp. 241-244, 1990.
Nishimura, T., Wake, A. and Fukumori, E., "Natural convection of water near the density extremum for a wide range of Rayleigh numbers, Numerical Heat Transfer (A), Vol. 27, pp. 433-449, 1995.
Ostrach, S., "Natural convection in enclosures," Adv. Heat Transfer, Vol. 8, pp. 161-227, 1972.
Ostrach, S., "Natural convection in enclosures," J. Heat Transfer, Vol. 110, pp. 1175-1190, 1988.
Paolucci, S. and Chenoweth, D. R., "Transient to chaos in a differentially heated vertical cavity," J. Fluid Mech., Vol. 201, pp. 379-410, 1989.
Patterson, J. C. and Imberger, J., "Unsteady natural convection in a rectangular cavity," J. Fluid Mech., Vol. 100, partl, Vol. 2, pp. 65-86, 1980.
Patterson, J. C. and Armfield, S. W., "Transient feature of natural convection in a cavity," J. Fluid Mech., Vol. 219, pp. 469-497, 1990.
Pecheux, J., Le Quere, P., and Abcha, F., "Curvature effects on axisymmetric instability of conduction regime in a tall air-filled annulus, Phys. Fluids, Vol. 6, No. 10, pp. 3247-3255, 1994.
Penot, F., N''Dame, A., and Le Quere, P., "Investigation of the route to turbulence in a differentially heated cavity," Proceedings of the ninth international Heat Transfer Conferences vol. 2, pp. 417-422, 1990.
Penot, F., N''Dame, A. and Le Quere, P., "Investigation of the route to turbulence in a differentially heated cavity," Heat Transfer Conf., Vol. 2, 1990.
Peutrec, Y. Le, and Lauriat, G., "Effects of the Heat Transfer at the Side Walls on Natural Convection in Cavities," J. Heat Transfer, vol. 112, pp. 370-378, 1990.
Prasad, V., and Kulacki, F. A., "Free convection heat transfer in a liquid-filled vertical annulus," J. Heat Transfer, vol. 107, pp. 596-602, 1985.
Robillard, L. and Vasseur, P., "Transient natural convection heat transfer on water with maximum density effect and supercooling," J. Heat Transfer, Vol. 103, pp. 528-534, 1981.
Robillard, L. and Vasseur , P., "Convective response of a mass of a mass of water near 4"C to a constant cooling rate applied on its boundaries," J. Fluid Mech., Vol. 118, pp. 123-141, 1982.
Ravi, M. R., Henkes, R. A. W. M., and Hoogendoorn, C. J., "On the high-Rayleigh-number structure of steady laminar natural-convection flow in a square enclosure," J. Fluid Mech. vol. 262, pp. 325-351, 1994
Patterson, J. C. and Imberger, J., "Unsteady natural convection in a rectangular cavity," J. Fluid Mech., Vol. 100, partl, Vol. 2, pp. 65-86, 1980.
Patterson, J. C. and Armfield, S. W., "Transient feature of natural convection in a cavity," J. Fluid Mech., Vol. 219, pp. 469-497, 1990.
Pecheux, J., Le Quere, P., and Abcha, F., "Curvature effects on axisymmetric instability of conduction regime in a tall air-filled annulus, Phys. Fluids, Vol. 6, No. 10, pp. 3247-3255, 1994.
Penot, F., N''Dame, A., and Le Quere, P., "Investigation of the route to turbulence in a differentially heated cavity," Proceedings of the ninth international Heat Transfer Conferences vol. 2, pp. 417-422, 1990.
Penot, F., N''Dame, A. and Le Quere, P., "Investigation of the route to turbulence in a differentially heated cavity," Heat Transfer Conf., Vol. 2, 1990.
Peutrec, Y. Le, and Lauriat, G., "Effects of the Heat Transfer at the Side Walls on Natural Convection in Cavities," J. Heat Transfer, vol. 112, pp. 370-378, 1990.
Prasad, V., and Kulacki, F. A., "Free convection heat transfer in a liquid-filled vertical annulus," J. Heat Transfer, vol. 107, pp. 596-602, 1985.
Robillard, L. and Vasseur, P., "Transient natural convection heat transfer on water with maximum density effect and supercooling," J. Heat Transfer, Vol. 103, pp. 528-534, 1981.
Robillard, L. and Vasseur , P., "Convective response of a mass of a mass of water near 4"C to a constant cooling rate applied on its boundaries," J. Fluid Mech., Vol. 118, pp. 123-141, 1982.
Ravi, M. R., Henkes, R. A. W. M., and Hoogendoorn, C. J., "On the high-Rayleigh-number structure of steady laminar natural-convection flow in a square enclosure," J. Fluid Mech. vol. 262, pp. 325-351, 1994
Vest, C. M., and Arpaci, V. S., "Stability of natural convection in a vertical slot," J. Fluid Mech. vol. 36, part I, pp. 1-15, 1969.
Vasseur, P. and Robillard, L., "Transient natural convection heat transfer in a mass of water cooled through 4"C," Int. J. Heat Mass Transfer, Vol. 23, pp. 1195-1205, 1980.
Vasseur, P. and Robillard, L. and Shekar, B. C., "Natural convection heat transferon water within a horizontal cylindrical annulus with density inversion effect," J. Heat Transfer, Vol. 105, pp. 117-123, 1983.
Watson, A., "The effect of the inverson temperature on the convection of water in an enclosed rectangular cavity," Q. J. Mech. Appl. Math., Vol. 25, pp. 423-446, 1972.
Wakitani, S., "Experiments on convective instability of large Prandtl number fluids in a vertical slot," J. Heat Transfer, vol. 116, pp. 120-126, 1994.
Weidman, P. D., and Mehrdadtehranfar, G., "Instability of natural convection in a tall vertical annulus," Phys. Fluids, Vol. 28, No. 3, pp. 776-787, 1985.
Xia, Q., Yang, K. T. and Mukutmoni, D., "Effect of imposed wall temperature oscillations on the stability of natural convection in a square enclosure", J. Heat Transfer, Vol. 117, pp. 113-120, 1995.
Xin, S., and Le Quere, P., "Direct numerical simulations of two-dimensional chaotic natural convection in a differentially heated cavity of aspect ratio 4," J. Fluid Mech. vol. 304, pp. 87-118, 19~5.