摘 要
本文以元素疊代（Element-by-Element）為基礎的投射有限元素流體解析法（projection finite element fluid analysis）為主，對於均勻紊流流過二維混合對流通道之無障礙物，具障礙物及三維強制對流通道之無障礙物，具障礙物的暫態流動及其傳熱現象進行分析。以EBE-PCG投射有限元素法取代以往有限元素法解無因次之Navier-Stokes Equation及能量方程式，乃是為了能減少矩陣的儲存量，及減少電腦上的計算時間。並且運用Large Eddy Simulation的紊流模式來模擬紊流流場，以前置處理之共軛梯度法（preconditioned conjugate gradient method）加以疊代而解出其速度場，壓力場和溫度場。
本文提供了一套EBE-PCG數值模擬二維和三維紊流流場的計算方法及結果，經由文獻的結果驗証，確認本計算方法的準確及可靠性。這些結果可以瞭解二維及三維通道內混合對流的紊流流場現象及熱傳效應。

ABSTRACT
This paper used Element-by-Element projection finite element fluid analysis turbulent flow and heat transfer for uniform turbulent flow in a 2D mixed convection channel and 3D force convection channel with heated block. Using Element-by-Element finite element preconditioned conjugate gradient method replaces the traditional finite element method for the non-dimensional Navier-Stokes Equation and Energy Equation. It can reduce matrix storage and computational time. Using large eddy simulation method turbulent model simulates turbulent flow with preconditioned conjugate gradient method and iterates to obtain velocity field, pressure field, and temperature field.
This paper offered a set of EBE-PCG numerical simulation method to simulate 2D and 3D turbulent flow. It can confirm that this method is precise and reliable by references. These conclusions can help us understand the phenomenon of 2D and 3D mixing convection of turbulent flow in the channel and heat transfer.

參考文獻
[1] T.J.R. Hughes, M. ASCE Itzhak Levit, and James Winget, ”Element-By-Element Implicit Algorithms for Heat Conduction,” Journal of Engineering Mechanics v109 n2 Apr 1983 p 576-585.
[2] T.J.R. Hughes, Itzhak Levit and James Winget, “An Element-By-Element Solution Algorithm For Problems of Structural and Solid Mechanics,” Computer Methods in Applied Mechanics and Engineering v36 n2 Feb 1983 p241-254 0374-2830.
[3] Miguel Ortiz, Peter M. Pinsky and Robert L. Taylor, “Unconditionally Stable Element-By-Element Algorithms for Dynamic Problem,” Computer Methods in Applied Mechanics and Engineering v36 n2 Feb 1983 p223-239 0374-2830.
[4] Thomas J.R. Hughes, James Winget, Itzhak Levit and T.E. Tezduyar, “New Alternate Directions Procedure in Finite Element Analysis Based Upon EBE Approximate Factorization,” Proceedings of the Symposium on Recent Developments in Computer Methods for Nonlinear Solid and Structural Mechanics, Proceedings of the ASME Joint Meeting of Fluid Engineering, Applied Mechanics and Bioengineering, University of Houston, Texas, June 1983.
[5] Thomas J.R. Hughes, A. Raefsky, A. Muller, J. Winget and I. Levit, “A Progress Report on EBE Solution Proceedings in Solid Mechanics,” Proceedings of the Second International Conference on Nonlinear Problems, Barcelona, Spain, April 1984.
[6] J.M.Winget, “Element by Element Solution Procedures for Nonlinear transient heat condition analysis,” Ph.D. Thesis, California Institute of Technology,1983.
[7] Itzhak Levit, “Element-By-Element Solvers of Order N,” Computers and Structures v27 n3 1987 p357-360.
[8] Graham F.Carey and Bo-nan Jiang, “Element-By-Element Linear and Nonlinear Solution Schemes,” Communications in Applied Numerical Methods, Vol 2 1986 p145-153.
[9] A.j. Wathen, ”An Analysis of Some Element-By-Element Techniques,” Computer Methods in Applied Mechanics and Engineering v74 1989 p271-287.
[10] Jocelyne Erhel, Alice Traynard and Marina Vidrascu, “An Element-By-Element Preconditioned Conjugate Gradient Method Implemented on Vector Computer,” Parallel Computing v17 1991 p1051-1065
[11] Manolis Papadrakakis and Michael C. Dracopoulos, “A Global Preconditioner for the Element-By-Element Solution Methods,” Computer Methods in Applied Mechanics and Engineering v88 1991 p275-286
[12] M.P. Reedy and J.N. Redy, “Penalty Finit Element Analysis of Incompressible Flows Using Element by Element Solution Algorithms,” Computer Methods in Applied Mechanics and Engineering v100 1992 p169-205
[13] Akira Mizukami, “Element-by-Element Penalty/Uzawa Formulation for Large Scale Flow Problems,” Computer Methods In Applied Mechanics and Engineering v112 1994 p283-289.
[14] Zhiping Li, M.B.Reed , ”Convergence Analysis for An Element-by-Element Finite Element Method,” Computer Methods in Applied Mechanics and Engineering V123 1995 P33-42.
[15] Adefemi Sunmonu, “Implementation of A Novel Element-BY-Element Finite Element Method on the Hypercube,” Computer Methods in Applied Mechanics and Engineering v123 1995 p43-51.
[16] M.P. Redy and J.N. Redy, “Multigrid Methods to Accelerate Convergence of Element-By-Element Solution Algorithms for Viscous Incompressible Flows,” Computer Methods in Applied Mechanics and Engineering v132 1996 P179-193.
[17] Y. Nakabayashi, H. Okuda and G.Yagawa, “Parallel Finite Element Fluid Analysis on An Element-By-Element Basis,” Computational Mechanics v18 1996 p377-382.
[18] A.J Chorin, “Numerical Solution of Navier-Stokes Equations,” Math. Comp., Vol 22 1968 pp.745-762.
[19] B. Ramaswamy, T.C. Jue and J.E. Akin, ”Semi-implicit and Explicit Finite Element Schemes for Coupled Fluid/Thermal Problems,” Int. J. Numer. Methods Eng. v34 1992 pp.675-696.
[20] T.J.R. Hughes, I. Levit and J.M. Winget, “Implicit, Unconditionally Stable Algorithms for Heat Conduction Analysis,” J. Eng. Mechanics Division ASCE v109 1983 p576-585.
[21] T.J.R. Hughes and J.M. Winget, ”Solution Algorithms for Nonlinear Transient Heat Conduction Analysis Employing Element by Element Iteratives Strategies,” Comput. Meth. Appl. Mechanics Eng. v52 1985 p711-815.
[22] Sungsu. Lee. ”Lager Eddy Simulation of Flow past a Square Cylinder Using Finite Element Method,”1997。

[23]林明德，雙模組電子元件散熱特性之探討，國立成功大學工程科學研究所碩士論文，1997。
[24]莊和達，運用EBE-PCG及投射有限元素法計算不可壓縮流體之熱傳現象，國立成功大學造船及船舶機械工程研究所碩士論文，2000。
[25]ATSUSHI OKAJIMA.”Strouhal numbers of rectangular cylinders,” J. Fluid Mech vol. 123. 379-398. 1982