臺灣博碩士論文加值系統

近十多年來，微機電系統領域已有顯著且令人印象深刻之發展。有關微流道方面，許多實驗與理論報告已經提出, 但仍然存在某些問題而具有研究之價值。本文利用數值計算來探討微管道中氣體流之行為。首先在流場區域建立非結構四面體與稜鏡型網格，然後採用四步Runge-Kutta時間積分與上風有限體積法求解卡氏座標系統下非穩態三維拿維-史托克方程式。為有效處理幾何之問題，將採用CATIA軟體建構外型及表面三角型與稜鏡型網格，然後利用Liu和Hwang所發展之非結構網格建立法完成四面體網格分佈。為探討低速流，本文使用Rossow所提出求解壓力之方法，在非結構四面體/稜鏡型網格上，本計算擴展此方法以求解三維非黏滯與層流。為評估此數值方法，首先探討球體與圓形截面之收斂-發散噴嘴之非黏滯流。在不同馬赫數下，球表面壓力係數及速度分佈與勢流理論比較並研究其收斂行為。至於噴嘴流，延噴嘴中心線之壓力和速度分佈與一維等熵數值解比較。其次進行圓管流計算。將結果與poiseuille 流之解析解(入流長度及poiseuille number)比較可知本數值方法在求解層流之準確性。最後將此求解步驟應用至梯形截面之微管道之研究。在不同努森數與不同馬赫數下計算Poiseuille number與壓力分布並與其他文獻之相關結果比較。

In the last decade, the microelectromechanical system(MEMS) field has significant and impressive progress. For the microchannel flows, a lot of experimental and theoretical results have been presented, but it still exists some problems which are worthwhile to study. In this thesis, the numerical computations are performed to investigate the gaseous flow behaviors in the microchannels. First the unstructured tetrahedral and prismatic meshes are created in the flow domain, the four-step Runge-Kutta time integration scheme and finite volume upwind method are adopted to solve the unsteady three-dimensional Navier-Stokes equations in he Cartesian coordinate system. To efficiently treat the geometric problem, the CATIA software is introduced to generate geometric shape, surface triangular grids and prismatic meshes. Then the mesh generation technique presented by Liu and Hwange is utilized to finish the distribution of tetrahedrons. To investigate the low speed flows, the approach for solving the pressure, which was presented by Rossow, is adopted in this study. On the unstructured tetrahedral/prismatic meshes, the above approach is extended to solve the three-dimensional inviscid and laminar flow. To evaluate this numerical method, the inviscid flows around sphere and passing through the converging-diverging nozzle with circular crosssection are investigated first. For the different Mach number, the comparision between the computed pressure coefficient and velocity distributions on the surface of sphere and the results from the potential flow theory is performed. Also the history of convergence is studied. About the nozzle flow, the pressure and velocity distributions along the nozzle axis are compared with those from the isentropic flow. Secondly, the computation of pipe flow is processed. For the comparision between the present results and the analytical solution for the poiseuille flow (such as entrance length and poiseuille number), the accuracy of current numerical approach for solving the laminar flow is confirmed. Finally, the present solution procedure is applied to study the trapezoidal microchannel. For the different value of Knudsen number and Mach number, the distribution of Poiseuille number and pressure distribution are computed and compared with the related data in the other literature.

中文摘要 I
Abstract III
ABSTRACT FOR EACH CHAPTER IN CHINESE V
Chapter I 序論 VI
Chapter II 數學模式及數值方法 VIII
Chapter III 網格建立 IX
Chapter IV結果與討論 X
Chapter V結論與建議 XI
致謝 XII
Outline XIII
List of Figure XV
Symbol description XVIII
I. Introduction 1
1.1 Motivation of the Study 1
1.2 Literature Survey 2
1.3 Content of This Thesis 5
II. Governing Equation and Numerical Approach 6
2.1 Governing Equation 6
2.2 Finite Volume Upwind Method 8
2.2.1 Inviscid Flux 8
2.2.2 Viscous Flux 12
2.3 Pressure Correction Equation 12
2.4 Time Integration and Accelerate Convergence 14
2.5 Boundary Condition 16
2.5.1 Boundary Condition for External Flow 16
2.5.2 Boundary Condition for Internal Flow 16
2.5.3 Solid Surface Boundary Condition 17
III Mesh Generation 19
3.1 Generation of Tetrahedrons for Sphere and Nozzle 20
3.2 Generation of Tetrahedral and Prismatic Meshes for Pipe and Trapezoidal Channel 20
IV Result and Discussion 22
4.1 Evaluation of the Revised Form of FDS Scheme 22
4.2 Validation of FDS Scheme with Pressure Correction 23
4.2.1 Inviscid Flow Over a Sphere 23
4.2.2 Applied Acceleration Scheme to Pressure Correction 24
4.2.3 Flow Over Sphere With Lower Speed 26
4.2.4 Inviscid Nozzle Flow 28
4.3 Viscous Pipe Flow 29
4.4 Micro-channel flow 30
V. Conclusion 33
Reference 36

1.Chorin, A. J., “A Numerical Method for Solving Incompressible Viscous Flow Problem,” Journal of Computational Physics, Vol. 2, 1967, pp. 12-26.
2.Steger, J. L. and Kutler, P., “Implicit Finite-Difference Procedures for the Computation of Vortex Wakes,” AIAA Journal, Vol. 15, April 1977, pp. 581-590.
3.Dochul Choi, Charles L. Merkle, “Application of Time-Iterative Schemes to Incompressible Flow,” AIAA Journal, Vol. 23, No. 10, 1985, pp. 1518-1524
4.B. van Leer, W.T. Lee and P.L. Roe, ”Characteristic Time-Steeping or Local Preconditioning of the Euler equations,” AIAA Paper, 91-1552, 1991
5.Turkel, E., “Review of Preconditioning Methods for Fluid Dynamics,” Applied Numerical Mathematics, 12, 1993, p.p. 257-284.
6.Turkel, E., Vatsa, V. N., and Radespiel, R., “Preconditionung Methods for Low-Speed Flows,” In preceedings of the 14th AIAA applied Aerodynamics Conference, AIAA Paper 96-2460(1996)
7.Ming-Zong Wang, “Solving Low-Mach Number Euler Flow with Local Preconditioning Technique,” the thesis of Master degree in NCKU, 2001.
8.Samet Y. Kadioglu, Mark Sussman, Stanley Osher, Joseph P. Wright, Myungjoo Kang, “A Second Order Primitive Preconditioner for Solving All Speed Multi-Phase Flows,” J. Compt. Phys., 209, 2005, p.p. 477-503.
9.Yi-Cheng Chen, “A Second Order Primitive Preconditioner for Solving All Speed Flows,” the thesis of Master degree in NCKU, 2006.
10.Herve Guillard, Cecile Viozat, “On the Behaviour of Upwind Schemes In the Low Mach Number Limit,” Computers and Fluids, 28, 1999, p.p. 63-86
11.C.-C. Rossow, “A Flux-Splitting Scheme for Compressible and Incompressible Flows,” J. Compt. Phys., 164, 2000, p.p. 104-122.
12.C.-C. Rossow, “A Blended Pressure/Density Based Method for the Computation of Incompressible and Compressible Flows,” J. Compt. Phys., 185, 2003, p.p. 375-398.
13.C.-C. Rossow, “Extension of a Compressible Code Toward the Incompressible Limite,” AIAA Journal, Vol. 41, No. 12, 2003, p.p. 2379-2386.
14.C. Y. Liu and C. J. Hwange, “New Strategy for Unstructured Mesh Generation,” AIAA Journal, Vol. 39, No. 6, June 2001, p.p. 1078-1085.
15.Yee-Yon Yong, “Computation of Viscous Flow Field and Noise for the Rotor-Blade on the Tetrahedral/ Prismatic Meshes,” the thesis of Master degree in NCKU, 2005.
16.Burden, R. L., Taires, J. D., Numerical Analysis, seventh Edition, California, Brooks/Cole, 2001.
17.James E. A. John, Gas dynamics, Second edition
18.Frank M. White , Viscous fluid flow, Second edition
19.G. L. Morini, M. Spiga, P. Tartarini, “The rarefraction effect on the friction factor of gas flow in microchannels,” Superlattices and Microstructures, 35(2004), 587-599
20.C.-C. Rossow, “Convergence Acceleration for Solving the Compressible Navier-Stokes Equations,” AIAA Journal, Vol. 44, No. 2, 2006, p.p. 345-352.
21.Chien-Wen Lai, “Generation of Dynamic Tetrahedral/Prismatic Meshes,” the thesis of Master degree in NCKU, 2005.
22.Guo-Wei Tang, “Generation of Tetrahedral/Prismatic Meshes and Its Applications in Microchannel Flows,” the thesis of Master degree in NCKU, 2006.
23.Huai-Gang Liu, “Numerical Study of Rarefaction and Compressibility Effects on Gas Flows in Microchannels,” the thesis of Master degree in NCKU, 2006.
24.Roe, P. L., “Approximate Riemann Solvers, Parameter Vectors ans Difference Schemes,” Journal of Compuatational Physics, Vol. 43, No. 2, 1981, p.p. 357-372.
25.M. S. Liou and C. J. Steffen, “A New Flux Splitting Scheme,” J. Comput. Phys., 107, 23, 1993.
26.R. C. Swanson, R.Radespiel, E. Turkel, “On Some Numerical Dissipation Schemes,” J. Comput. Phys., 147, 518, 1998.
27.R. C. Swanson, E. Turkel, “Multistage Schemes With Multigrid for Euler and Navier-Stokes Equations,” NASA Technical Paper 3631, 1997.
28.Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, Berlin, Springer, New York, 1999.
29.T. J. Chung, Computational fluid dynamics, Cambridge University Press, 2002