臺灣博碩士論文加值系統

摘 要

本文在探討高普蘭特數(Pr)的變黏度流體在兩水平同心及偏心圓球之自然對流問題。並與定黏度流體比較，得知變黏度對於流場、溫度場及熱傳率的效應皆有明顯的影響。流場隨著雷利數之增加，由原來的單一對流渦漩崩潰而產生不穩定之二級渦漩及反向旋轉的三級渦漩；變黏度使得流體溫度梯度變大，增強熱對流效果而提高熱傳率。黏度與溫度之關係是使用四個參數型式之關係式來模擬，格點系統則使用一組與邊界近乎正交之格線配合權函數法；所用的數值方法係以連續鬆馳法（SOR）解流線方程式，並以交錯直接隱性法（ADI）解渦漩及溫度場。為確保數值方法之精確性，以定黏度牛頓流體分析結果和參考文獻作比較後，再分析變黏度流體之熱傳特性。
分析結果得知流場隨著雷利數(Rayleigh Number ,Ra )增加，會增強對流效果而提高熱傳率；正偏心為外圓球向上提升其幾合結構可得較佳之熱傳效果，相反地負偏心的幾何結構較不利於自然對流的發展空間，使得熱傳效果較低。

ABSTRACT
A numerical simulation has been carried out for the natural convection of high Prandtl number fluids with temperature-dependent viscosity between concentric and vertically eccentric spheres. Furthermore, the results are used to compare with the cases for constant viscosity and the comparison shows that the variable viscosity has the influence on flow field, temperature distribution and heat transfer. Besides, when Rayleigh number is larger than the first critical value, the unicellular convection vortex is broken into a set of unsteady secondary vortices. For Rayleigh number larger than the second critical Rayleigh number, the two secondary vortices re-broken into some smaller counter-rotating tertiary vortices. Temperature gradient increases with variable viscosity, so this can enhance the convection and heat transfer of the flow field, and let the flow become unstable. A four-parameter correlation for the variable viscosity is adopted. When the modified Sorenson's method used to generate the grid line, it can get the grid system with orthogonality along all boundaries and this system can enhance calculation accuracy. The grid system goes along with weighting function scheme (WFS) to discrete the general governing equation. The successive over-relaxation method (SOR) is applied to solve stream function equation and the alternating direction implicit method (ADI) is applied to solve vorticity and energy equations. For giving one confidence to the numerical method in this thesis, the numerical results for constant viscosity are used to compared with the provious experimental results, and have good agreements with the previous.
The analysis shows that the convection becomes more obvious and heat transfer increases with increasing Rayleigh number. Moreover, the influence of eccentricity on the flow field is also studied in this thesis. It reveals that the positive eccentricity can improve the average Nusselt number, but the negative eccentricity may not promote the convection effect.

目錄
摘要…………………………………………………………………………..I
目錄…………………………………………………………………………..II
表目錄……………………………………………………………………….III
圖目錄………………………………………………………………………. V
符號說明…………………………………………………………………..VIII
第一章 緒論
1-1研究目的及背景…………………………………………………….1
1-2文獻回顧…………………………………………………………….3
1-3本文研究方法及結構……………………………………………….7
第二章 變黏度關係式
2-1黏度─溫度關係式………………………………………………….8
2-2甘油水容液………………………………………………………...10
第三章 理論分析
3-1物理模型及基本假設……………………………………………...13
3-2統制方程式之建立………………………………………………...14
第四章 數值分析
4-1前言………………………………………………………………...19
4-2計算平面上之統制方程式………………………………………...21
4-3數值解法…………………………………………………………...25
4-4計算流程及收歛準則……………………………………………...28
4-5物理量計算………………………………………………………...30
第五章 結果與討論
5-1程式測試………………………………………………………….36
5-2水平同心圓球暫態之自然對流……………….……37
5-3垂直偏心圓球暫態之自然對流…….……………39
5-4水平同心圓球穩態之自然對流…………………….41
5-5垂直偏心圓球穩態之自然對流…………………….45
第六章 結論…………………………………………………………….80
參考文獻……………………………………………………………………81
附件1……………………………………………………………………….89

參考文獻
1Elder, J. W., "Laminar free convection in a vertical slot," J. Fluid Mech. Vol.23, pp.77-98, 1965."
2Vest, C. M., and Arpaci, V. S., "Stability of natural convection in a vertical slot," J. Fluid Mech. Vol.36, pp.1-15, 1969.
3 Hart, J. E., "Stability of the flow in a differentially heated inclined box," J. Fluid Mech. Vol.47, pp.547-576, 1971.
4Chen, Y. M., and Pearlstein, A. J., "Stability of free convection flows of variable-viscosity fluids in vertical and inclined slots," J. Fluid Mech. Vol.198, pp.513-541, 1989.
5Seki, N., Fukusako, S., and Inaba, H., "Visual observation of natural convective flow in a narrow vertical cavity," J. Fluid Mech. Vol.84, pp.695-704, 1978.
6Chen, 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.
7Thangam, S. and Chen, C. F., "Stability analysis on the convection of a variable viscosity fluid in an infinite vertical slot," Phys. Fluids, Vol.29, pp.1367-1372, 1986.
8Chen, Y. M., and Pearlstein, A. J., "Viscosity-temperature correlation for glycerol-water solutions," Ind. Engng. Chem. Res. Vol.26, pp.1670-1672, 1987.
9Gill, A. E. and Kirkham, C. C., "A note on the Stability of convection in a vertical slot," J. Fluid Mech. Vol.42, pp.125-127, 1970.
10Chen, F. L. and Wu, C. H., "Unsteady Convective flows in a vertical slot containing variable viscosity fluids," Int. J. Heat Mass Transfer, Vol.36, pp.4233-4246, 1993.
11Jin, Y. Y. and Chen, C.F., "Natural Convection of high Prandtl number fluids with variable viscosity in a vertical slot," Int. J. Heat Mass Transfer 39, pp.2663-2670, 1996.
12Gavis, J. and Laurence, R. L., "Viscous heating in plane and circular flow between moving surfaces," I&EC Fundamentals, Vol.7, pp.232-239, 1968.
13Bottaro, A., Metzener, P. and Matalon, M., "Onset and two-dimensional patterns of convection with strongly temperature-dependent viscosity," Phys.Fluids A, Vol.4, No.4, pp.655-663, 1992.
14Shannon, R. L. and Depew, C. A., "Force laminar flow convection in a horizontal tube with variable viscosity and free convection effect," ASME J. of Heat Transfer, Vol.91, pp.251-258, 1969.
15Ockendon, M. and Ockendon, J. R., " Variable-viscosity flows in heated and cooled channels," J. of Fluid Mech., Vol.83 pp.177-190, 1977.
16Fujii, T., Takeuchi, M., Fujii, M., Suzaki, K. and Uehara, H., "Experiments on nature convection heat transfer from the outer surface of a vertical cylinder to liquids," Int. J. Heat Mass Transfer, Vol.13, pp.753-787, 1970.
17Jang, J.Y. and Lin, C. N., "The laminar free convection of a liquid with variable viscosity," J. of CSME, vol.8, No.3, pp.165-171, 1987.
18Jang, J.Y. and Lin, C. N., "Free convection flow over an uniform heat flux surface with temperature-dependent viscosity," Thermo and Fluid Dynamic, Vol.23, pp.213-217, 1988.
19Kassoy, D. R. and Zebib, A., " Variable viscosity effects on the onset of convection in a porous media," The Physics of Fluids, Vol.18, pp.1649-1651, 1975.
20Horne, R. N. and O’Sullivan M. J., "Convection in a porous medium heated from below: the effect of temperature dependent viscosity and thermal expansion coefficient," J. Heat Transfer, Vol.100, pp.448-452, 1978.
21Wazzan, A. R., Okamura, T. T. and Smth, A. M. O., "The stability of water flow over heatdt and cooled flat planes," J. Heat Transfer, Vol.90, pp.109-114, 1968.
22Wazzan, A. R., Okamura, T. T. and Smth. A. M. O., "The stability and transition of heated and cooled incompressible laminar boundary layers," Proc. Fourth Inter. Heat Transfer Conf. Paris, 1970.
23Strazisar, A., Prahl, J. M. and Reshotko,E., "Experimental study of the stability of heated laminar boundary layers in water," Report FTAS/TR 75-113, Dept of Fluid Thermal and Aero. Sci., Case Western Reserve Univ., 1975.
24Jang, J. Y. and Mollendorf, J. C., "The Stability of a vertical natural convection boundary layer with temperature dependent viscosity," Int. J. Engng. Sci., Vol.26, pp.1-12, 1988.
25Leu, J. S. and Jang, J. Y., "Variable Viscosity and non-Darcian effects on the flow and vortex instability of natural convection boundary layer flows," Ph. D. Thesis, National Cheng Kung University, 1994.
26莊榮南, “變黏度流體在同心及偏心圓環間自然對流研究”, 國立成功大學機械工程研究所博士論文,1998.
27周煥銘, "應用座標轉換法探討微極流體之自然對流熱傳問題研究",國立成功大學機械工程研究所博士論文, 1993.
28Anderson, D.A.,Tannehill J. C., and Pletcher, R. H., “Computation Fluid Mechanics and Heat Transfer,” Hemisphere New York, Chap. 10, pp. 519-546, 1984.
29Thompson, J. F., Thames, F. C., and Mastin, C. W., “Automatic Numerical Generation Of Body-fitted Curvilinear Coordinate System for Field Containing and Number of Arbitrary Two-Dimensinal Bodies,” Journal of Computational Physice, vol.15, pp. 299-319,1974.
30Thmmes, F. C., Thompson, J. F., and Mastin, C. W., “Numerical Solution of the Navier-Stokes Equations for Arbitrary Two-Dimensional Airfoils,” Langley Research Center, NASA, pp. 347,1975.
31Thompson, J. F., Thames, F. C., and Mastin, C. W., “Boundary-Fitted Curvilinear Coordinate System for Solution of Partial Differential Equations of fields Containing Amy Number of Arbitrary Two-Dimensional Bodies,” NASA CR-2729, 1976
32Thames, Frank C., Thompson, Joe F., Mastin, c. W., and Wacker, R., “Numerical Solution for Viscous and Potential Flow about arbitraray Two-Dimensional Bodies Using Body-Fitted Coordinate System,” Journal of computational Physics, Vol. 24, pp. 245-273,1977.
33Thompson, J. F., Thames, F. C. and Mastin, C. W., “TOMCAT-A Code for Numerical Generation of Boundary-Fitted Curvilinear Coordinate Systems on Fields Containing Any Number of Arbitrary Two-Dimensional Bodies,” Journal of Computational Physics, vol. 24, pp. 274-302, 1977.
34Thomas, P. D. and Middlecoeff, J. F., “direct control of the Grid Point Distribution in Meshes Generated by Elliptic Equations,” AIAA Journal, vol. 18, No. 16, pp. 652-656, 1980.
35Thomas, P. D., “Composite Three-dimensional Grids Generated by Elliptic Systems,” AIAA Journal, vol. 20, No. 9, pp. 1195-1202, September, 1982.
36Thames, F. C., “Generation of 3-D Boundary-Fitted Curvilinear coordinate system for Wing/Wing Tip Geometries Using the Elliptic Solver Method,” Numerical Grid Generation, Thompson, J. F. ed., pp.695-716,1992.
37Jeng, Y. N. and Liou, Y. C., “Two Modified Versions of Hsu-Leels Elliptic Slover of Grid Generation, “Numerical Heat Transfer, Part B, Vol. 22, pp. 125-140,1992.
38Marcel, V., “On One-dimensional Stretching Functions for finite-Difference Calculations,” Journal of Comptational Physics, Vol. 50, pp.215, 1983.
39Dwyer, H. A., Kee, R. J., and Sanders, B. R., “Adaptive Grid Method for Problems in Fluid Mechanics and Heat Transfer,” AIAA J., Vol. 18, No.10, pp. 1205-1212, 1980.
40Dwyer, H. A., “Grid Adaptive for Problems in Fluid Dynamics,”AIAA J., Vol. 22, No. 12, pp. 1705-1212, 1984
41Shyy, W., “An Adaptive Grid Method for Navier-Stokes Flow Computation: Grid Addition, “Applied Numerical Mathematics, Vol. 2, pp. 9-19, 1986.
42Shyy, W., “An Adaptive Grid Method for Navier-Stokes Flow Computation,” Applied Mathematics and Computation, Vol. 21, pp. 201-209, 1987
43Shyy, W., “A Numerical Study of Annular Dump diffuer Flows,” Comput. Methods Appl. Mech. Eng., Vol. 53, pp. 47-65,1985.
44Jeng, Y. N. and Liou, S. C., “Modified Multiple One Dimensional Adaptive Grid Method,” Numerical Heat Transfer, Part B, Vol. 15, pp. 241-247,1989.
45Jneg, Y. N. and Liou, Y. C., “A New Adaptive Grid Generation by Elliptic Equations with Orthogonality at All of the Boundaries,” Journal of Scientific Computing, Vol. 7. No. 1, 1992.
46Lee, D. and Tsuei, Y. M., “A Modified Adaptive Grid Method for Recirculating Flows,” Int. Journal for Numerical Methods in Fluids, Vol. 14,pp. 775-791,1992.
47Stengel, K. C., Oliver, D. S. and Booker, J. R., "Onset of convection in a variable-viscosity fluid," J. Fluid Mech. Vol.120, pp.411-431, 1982.
48Segur, J. B., "Physical properties of glycerol and its solutions," In Glycerol (ed. C. S. Miner & N. N. Dalton), pp.238-334, Reinhold, 1953.
49Segur, J. B. and Oberstar, H. E., "Viscosity of glycerol and its aqueous solutions," Ind. Engng. Chem. Vol.43, pp.2117-2120, 1951.
50Badr, H. M., "Study of laminar free convection between two eccentric horizontal tubes," Trans. of the CSME, Vol.7, No.4, 1983.
51Cho, C. H., Chang, K. S. and Park, K. H., "Numerical simulation of natural convection in concentric and eccentric horizontal cylindrical annuli," ASME J. Heat Transfer, Vol.104, pp.624-630, 1982.
52Projahn, V., Reiger, H. and Beer, H. " Numerical analysis of laminar convection between concentric and eccentric cylinders," Numerical Heat Transfer, Vol.4, pp.131-146, 1981
53Prusa, J. and Yao, L. S., "Natural convection heat transfer between eccentric horizontal cylinders," ASME J. Heat Transfer, Vol.105, pp.108-116, 1983.
54陳玟瑞, "同心與垂直偏心圓球內之暫態之自然對流熱傳研究",國立成功大學機械工程研究所博士論文, 1995.
55Yin, S. H., Powe, R.E., Scanlan, J. A., and Bishop, E.H., “Natural convection flow patterns in spherical annuli,” Int. J. Heat Mass Transfer 16, 1785-1795(1973).
56Caltagirone, J. P., Combarnous, M., and Mojtabi, A., “Natural convection between two concentric spheres: transition toward a muticellular flow,” Heat Mass Transfer 3, 107-114(1980).
57Lee, S. L., Chen, T. S., and Aramaly, B. F., "New finite difference solution methods for wave instability problem," Numerical Heat Transfer, Vol.10, pp.1-18, 1986.
58Lee, S. L., Chen, T. S., and Aramaly, B. F., "Nonparallel wave instability analysis of boundary layer flows," Numerical Heat Transfer, Vol.12, pp.349-366, 1987.
59Lee, S. L., "A new numerical formulation for parabolic differential equations under the consideration of large time steps," Int. J. for Numerical Methods in Engineering, Vol.26, pp.1541-1549, 1988.
60Lee, S. L., "Weighting function scheme and it’s application on multi-dimensional conservations, " Int. J. Heat Mass Transfer, Vol.32, pp.2065-2073, 1987.
61Fujii, T., Honda, T., and Fujii, M. “A numerical analysis of laminar free convection around an isothermal sphere (effects of space and Prandtl number),” Proc. 1987 ASME-JSME Thermal Engineering Joint Conference,Vol.4,pp. 55-60(1987)
62Ozoe, H., Fujii, K., Shibata, T., Kuriyama, H., and Churchill, S. W., “Three-dimensional numerical analysis of natural convection in a spherical annulus,” Numerical Heat Transfer 8,383-406(1985).
63Ozoe, H., Kuriyama, H., and Takami, A., “Transient natural convection in a spherical and a hemispherical enclosure,” Proc. 1987 ASME-JSME Thermal engineering Engineering Joint Conference, Vol. 4, pp. 19-25(1987)
64Bishop, E. H., Carley, C. T., and Powe, R. E., “Natural Convection Flow Patterms in Cylindrical Annuli,” Int. j. Heat Mass Transfer, 11, pp.1741-1752,1968.
65Kuehn, T. H., and Goldstein, R. J.,” An experimental and theoretical Study of Natural in the Annulus Between Horizontal Concentric Cylinders,” J. Fluid Mech., Vol. 74,pp. 695-719,1976
66Caltagirone, J. P., Combarnous, M., and Mojtabi, A., “Natural convection between two concentric spheres: transition toward a muticellular folw,” Numerical Heat Transfer 3, 107-739(1980).