本文研究的目的在於利用數值方法模擬在三維封閉矩形內的自然對流與熱傳導，數值方法是以上風的有限體積法來求解三維Navier-Stokes equation，在連續方程式中加入人工壓縮因子（artificial compressibility）及壓力對時間的微分項，其中對流項採用高階的上風有限體積法，進而利用DDADI數值法對時間積分，並加入隱式殘值平滑性（implicit residual smoothing）加速穩態計算的收斂性。
在物理模型方面應用於一導熱基板上鑲有三塊通有一均勻熱通量並充滿工作流體為空氣的三維封閉矩形，晶片用不同的排列方式鑲於傳導基板上(垂直、水平與平躺)。主要模擬雷利數、基板與晶片的熱傳導係數對三維封閉矩形內的自然對流與晶片和基板間熱效應的影響。

A numerical method is developed to investigate the three-dimensional conjugate heat transfer of natural convection and conduction problems in a cubic enclosure. The Rayleigh number is selected between 10 to10 . The numerical method applied a finite-volume technique to solve the three -dimensional steady Navier-Stokes equations. The artificial compressible method, a third-order upwind finite volume method, DDADI time integration and an implicit residual smoothing were applied in the numerical method to achieve a higher-order accurate method.
In this article, we are concerned with the three-dimensional iterative- coupled heat conduction-convection problem for three heated chips mounted on a conductive substrate in a cubic enclosure filled with air. The chips are assumed to have uniform properties, with internal heat generation rate of Q [W], are mounted by different arrangements (Vertical, Horizontal, and Bottom). The main objective of the present work is to simulate the effect of Rayleigh number and substrate thermal conductivity on the three-dimensional natural convection flow within the cubic and on the heat transfer from the chips.

1. Davis V., 1983, “Natural Convection of Air in a square Cavity: A Bench mark Numerical Solution,” International Journal Numerical Methods Fluids, Vol. 3, pp. 249-264.

2. Markatos, N. C. and Pericleous, K.A., 1984, “Laminar and Turbulent Natural Convection in an Enclosed Cavity,” International Journal of Heat and Mass Transfer, Vol. 27, pp. 755-772.

3. Keyhani, M., and Chen, L., 1994, ”The Aspect Ratio Effect on Natural Convection in an Enclosure with Protruding Heat Sources,” Transactions of the ASME, Vol. 113, pp. 883-891.

4. Ho, C.J., and Chang, J. Y., 1994, ”A study of Natural Convection Heat Transfer in A Vertical Rectangular Enclosure with Two-Dimensional Discrete Heating: Effect of Aspect Ratio,” International Journal of Heat and Mass Transfer, Vol. 37, pp. 917-925.

5. House, J. M., and Christoph Beckermann, 1990, ”Effect of A Centered Conducting Body on Natural Convection Heat Transfer in an Enclosure,” Numerical Heat Transfer, Part A, Vol. 18.

6. Pallaers, J., Cuesta, I., Gran, F. X., and Giralt, F., 1995, ”Natural Convection in a Cubical Cavity Heated from Below at Low Rayleigh Numbers,” International Journal of Heat and Mass Transfer, Vol. 39, pp. 3233-3247.

7. Wroblewski, D. E. and Joshi, Y., 1992, ”Transient Natural Convection From a Leadless Chip Carrier in a Liquid Filled Enclosure: A Numerical Study,” Journal of Electronic Packaging, Vol. 114.

8. Wroblewski, D. E. and Joshi, Y., 1993, “Computations of liquid immersion cooling for a protruding heat source in a cubical enclosure,” International Journal of Heat and Mass Transfer, Vol. 36, No. 1201-1218.

9. Wroblewski, D. E. and Joshi, Y., 1994, “ Liquid Immersion Cooling of a Substrate Mounted Protrusion in a Three-Dimensional Enclosure: The Effects of Geometry and Boundary Conditions,” Transactions of the ASME, Vol. 116, February.

10. Wroblewski, D. E. and Joshi, Y., 1995, “Computations for a Three-by-Three Array of Protrusions Cooled by Liquid Immersion: Effect of Substrate Thermal Conductivity,” Transactions of the ASME, Vol. 117, December.

11. Adams, V. H. and Joshi, Y., 1999, ”Three-Dimensional Study of Combined Conduction, Radiation, and Natural Convection From Discrete Heat Sources in a Horizontal Narrow-Aspect-Ratio Enclosure, ” Transactions of the ASME, Vol. 121, November.

12. Liu, Y., Kemp, R. and XiaoLinLuo, 1997,“There-Dimensional Coupled Conduction-Convection problem for Three Chips Mounted on a Substrate in an enclosure,” Numerical Heat Transfer, Part A, Vol. 32.

13. Tou, S. K., Tso, C. P., and Zhang, X., 1999, “3-D Numerical Analysis of Natural Convective Liquid Cooling of a 3x3 Heater Array in Rectangular Enclosures,” International Journal of Heat and Mass Transfer, Vol. 42, No. 3231-3244.

14. Sezai, I. and Mohamad, A. A., 1999, ”Natural Convection from a Discrete Heat Source on the Bottom of a Horizontal Enclosure,” International Journal of Heat and Mass Transfer, Vol. 43. No. 2257-2266.

15. Ha, M. Y. and Jung, M. J., 2000, ”A Numerical Study on Three- Dimensional Conjugate Heat Transfer of Natural Convection and Conduction in A Cubic Conducting Body,” International Journal of Heat and Mass Transfer, Vol. 43. No. 4229-4248.

16. Chorin, A. J., 1967, ”A Numerical Method For Solving Incompressible Viscous Flow Problem,” J. Compute. Phys. Vol. 22, pp. 12-26.

17. Kloper, G .H. and Hung, C. M., ”A Diagonalized Diagonal Dominant Direction Implicit (D3ADU) Scheme and Sub iteration Correction,” AIAA Paper 98-2824.

18. MacCormack, R.W., ”A New Implicit Algorithm for Fluid Flow,” AIAA Paper CP-97-2100.

19. Bardina, J. and Lombard, C. K., ”Three Dimensional Hypersonic Flow Simulations with The CSCM Implicit Upwind Navier-Stokes Method,” AIAA Paper 87-1114.

20. Lin, S. Y. and Yu, Z. X., 2001, ”Finite Volume Scheme for Incompressible Flows and it’s Application to Internal External Flows,” Institute of Aeronautics and Astronautics National Cheng Kung University, Taiwan, R.O.C., October.

21. Osher, S. and Chakravarthy, S. R., 1984, “Very High Order Accurate TVD Scheme,” ICASE Report No. 44-84, 1985, also The IMA Volumes in Mathematics and its Applications, Vol. 2, Springer-Verlag, pp. 229-274.

22. Yeh, D. Y., Unsteady Aerodynamic and Aero-elastic Behaviors of Acoustically Excited Transonic Flow, Dissertation for Doctor of Philosophy Institute of Aeronautics and Astronautics National Cheng Hung University, Taiwan, R.O.C.

23. Fusegi, T., Hyun, J.M., Kuwahara, K., and Farouk B., 1991, “A Numerical Study of Three-Dimensional Natural Convection in a Differentially Heated Cubical Electronic,” International Journal of Heat and Mass Transfer, Vol. 34, pp. 1543-1557.

24. Krane, R. J. and Jessee, J., 1986, ”Some Detailed Field Measurements for a Natural Convection Flow in a Vertical Square Enclosure,” Proc. 1st, ASME-JSME Thermal Engng Joint Conf., Vol. 1, pp. 323-329.

25. Bilski, S. M., Loyd, J. R., and Yang, K. T. 1990, ”An Experimental Investigation of The Laminar Natural Convection Velocity in Square and Partitioned Enclosures,” ASME, Journal of Heat Transfer, Vol. 112, pp. 370-378.

26. Peutrec, Y. L. and Lauriat, G., 1990, ” Effects of The Transfer at The Side Walls on Natural Convection in Cavities,” ASME, Journal of Heat Transfer, Vol. 112, pp. 370-378.

27. Shunichi Wakitani, 2001, ”Numerical Study of Three-Dimensional Oscillatory Natural Convection at Low Prandtl Number in Rectangular Enclosures,” ASME, Journal of Heat Transfer, Vol. 123, pp. 77-83.

28. Leong, W. H. and Brunger, A. P., 1999, “Experimental Nusselt number for a cubic-cavity benchmark problem in natural convection,” International Journal of Heat and Mass Transfer, Vol. 42.

29. Raithby, G. D. and Wong, H.H., 1981, “Heat Transfer by Natural Convection Across Vertical Air Layers,” Numerical Heat Transfer, Part A, Vol. 04.

30. Quere, P. L., 1991, “Accurate Solutions to The Square Thermally Driven Cavity at High Rayleigh Number,” Computers Fluids, Vol. 20, No. 1.