臺灣博碩士論文加值系統

本文旨在使用局部預調法(Local Preconditioning Method)來解低馬赫數的尤拉方程式，使原本能解可壓縮流(Compressible Flow)的數值模式，可同時模擬不可壓縮流(Incompressible Flow)。首先，我們由原始變數 的三維尤拉方程式，導出其Flux Jacobin矩陣 ，然後乘以一預調矩陣 ，再利用矩陣轉換 ，使其變為保守變數 預調化後的Flux Jacobin矩陣 。局部預調法主要是在改變尤拉方程式的數值模式，使其在低馬赫數時，能克服局部聲速 ，和局部速度 差距過大，所導致收斂困難和解失真等的問題。
在數值方法上，我們則是使用網格中心有限體積上風法( Cell Centered Finite Volume Upwind Method)，將我們所要探討的物理區域，分割成無數個不規則的四面體，至於在通量的計算上，則以Roe所提的通量差分分離法(Flux Difference Splitting)來達到上風近似，此外為達高階準確上風法，更以Frink所提的重建算則，使其空間離散準確度由一階提高到二階。至於在時間積分方面則是採用四階的Runge-Kutta 方法，為了加速收斂，並使用了局部時階法(Local Time Stepping)，與殘值平滑法(Residual Smoothing)等技巧。最後由3D的RAE Wing-Body的收斂圖，和壓力圖(CP)得到證明，經過預調化後的保守尤拉方程式，有較準確的數值解。在本論文中我們所使用的計算器分別為CPU=AMD Athlon 800 ; RAM= 640MB 和CPU=PentiumⅢ800；RAM=768MB兩種。

The aim of this paper is using local preconditioning methods to solve nearly incompressible flow problems with numerical algorithms that were designed for compressible flow。First，we calculate the flux jacobin of primitive variables in 3D Euler equations in the Cartesian coordinate system，then multiply a preconditioner ，and transfer matrices after preconditioning。Final，we obtain the flux jacobin matrix of conservative variables 。The aim of the local preconditioning is changing numerical modes of Euler equations，and overcome the large disparity of the acoustic wave speed ，and the convected waves at the fluid speed 。The preconditioning that are applied here not only accelerate the convergence to a steady state but also change steady-state solution。
In numerical methods，we use cell-centered finite volume upwind method，and the inviscid flux vector is computed by Roe’s flux difference splitting 。In order to raise the level of the space difference scheme，we use Frink cell reconstruction schemes。The four-stage Runge-Kutta scheme is used to achieve the time integration。To accelerate the convergence，local time stepping and residual smoothing are utilized。Finally，we can endvince that the conservative Euler equation after preconditioning have better pressure contours diagrams。

目錄
中文摘要
英文摘要
誌謝
目錄
表目錄
圖目錄
符號說明
第一章緒論
1.1研究動機與目標
1.2預調算法的原理
1.3文獻回顧
1.3.1常見的預調算子
1.3.2預調算子的比較
1.3.3預調算子的功用
1.3.4 Condition Number的討論
1.3.5理想預調算子的設計
1.3.6預調算子的優點
第二章尤拉流場解和預調算法
2.1三維非保守尤拉方程式
2.2預調算法
2.3三維保守尤拉方程式
2.4矩陣轉換
第三章數值方法
3.1有限體積法
3.2 Roe’s通量差分離法
3.3重建算則法
3.4時間積分法
3.5穩態流場之加速收斂法
3.6邊界條件
第四章結果與討論
4.1程式結果的驗證
4.2預調矩陣參數的討論
4.3結果與討論
第五章結論與建議
5.1結論與建議
參考文獻

參考文獻
[1] Turkel, E., Vatsa, V. N., and Radespiel, R., “Preconditioning Methods for Low-Speed Flow,” In Proceedings of the 14th AIAA Applied Aerodynamics Conference, AIAA Paper 96-2460 (1996).
[2] Turkel, E., “A Review of Preconditioning Methods for Fluid Dynamics,” Applied Numerical Mathematics , Vol. 12, pp. 257-284 (1993).
[3] Turkel, E., ”Preconditioned Methods for Solving the Incompressible and Low Speed Compressible Equations,” Journal of Computational Physics, Vol. 72, pp. 277-298 (1987).
[4] Darmofal, D. L. and Schmid, P. J. “The Importance of Eigenvectors for Local Preconditioners of the Euler Equations,” 12th AIAA Computational
Fluid Dynamics Conference, pp. 102-117 (1995).
[5] Hsu, C. A., “三維不可壓縮黏性流數值模式發展與應用,” 中興工程季刊, 第六十期, pp. 1-37 (1998).
[6] Hsu, C. A., “可同時解不可壓縮流及可壓縮流之數值模式發展,” 中興工程季刊, 第六十二期, pp. 11-34 (1999).
[7] Y. S. Wang and H. Li., “A Preconditioning Techniques for Steady Euler Solutions ,” Int. J. Numer. Meth. Fluids 30, pp. 541-555 (1999).
[8] Gunther Brenner and Nafiz Alemdaroglu “Improvement to Compressible Euler Methods for Low-Mach Number Flows,” Int. J. Numer. Meth.Fluids
34, pp. 167-185 (2000).
[9] Hirsh, R. S., ”Higher Order Accurate Difference Solutions of Fluid Mechanics Problem by a Compact Differencing Technique ,” J. Comput. Phy. Vol. 19, pp. 90-109(1975).
[10] Partaker, S. V., ”Numerical Heat Transfer and Fluid Flows,” McGraw- Hill. New York. (1980).
[11] Chorine, A. J., ”A Numerical Method for Solving IncompressibleViscous
Flow Problem,” J. Compt. Phys., Vol.2, pp. 12-26(1967).
[12] Dohyung Lee, ”Design Criteria for Local Euler Preconditioning,” Journal of Computational Physics 144,pp. 423-459(1998).
[13] Dohyung Lee,”Local Preconditioning of the Euler and Navier-Stoke Equations,” P h. D. Thesis in The University of Michigan (1996).
[14] Jameson A, Schmidt W, Turkel E., ”Numerical Solution of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time Steeping Schemes,” AIAA Journal, pp. 81-125 (1981).
[15] Barth, T. J., ” The Design and Application of Upwind Scheme on Unstructured Meshes,” AIAA, Paper. 89-0366. (1989).
[16] Barth, T. J.,” A 3-D Upwind Euler Solver for Unstructured Meshes,” AIAA Paper. 91-1548-CP (1991).
[17] Roe, P. L., ”Approximate Riemann Solvers, Parameter Vectors, and Difference Scheme,” Journal of Computational Physics, Vol.43, pp. 357-372 (1981).
[18] Frink, N. T., Parikh, P., and Pirzadeh, S.,”A Fast Upwind Solver for the Euler Equations on Three-Dimensional Unstructured Meshes,” AIAA Paper. 91-0102 (1991).
[19] Frink, N. T., ”Upwind Scheme for Solving the Euler Equations on Unstructured Tetrahedral Meshes,” AIAA Journal, Vol.30, No.1, pp. 70-77 (1992).
[20] Yeu-Ching Perng and Ruey-Hor Yen ”Implicit Weighted Eno Schemes for Two and Three Dimensional Navier-Stokes Equations,” the six National Computational Fluid Dynamics Conference, pp. 385-392 (1997).
[21] Treadgold, D. A., Jones, A. F., and Wilson, K. H., “Pressure Distribution Measured in the RAE 8ft×6ft Transonic Wind Tunnel on RAE Wing ‘A’ in combination with an Axis-symmetric Boday at the Mach Number of 0.4,0.8,and0.9,”Experimental Data Base for Computer Program Assessment, Appendix B4, AGARD-AR-138 (1979).
[22] H.Cabuk, “Explicit Runge-Kutta Method for Three-Dimensional Internal Incompressible Flows,” AIAA Journal, Vol. 30, No. 8, August (1992).