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系統識別號 U0026-0812200914083488
論文名稱(中文) 球面活塞問題中自相似解之探討
論文名稱(英文) The Self-Similar Flow of the Spherical Piston Problem
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 96
學期 2
出版年 97
研究生(中文) 林冠廷
研究生(英文) Kuan-Ting Lin
學號 l1694113
學位類別 碩士
語文別 英文
論文頁數 50頁
口試委員 口試委員-沈士育
口試委員-王辰樹
指導教授-連文璟
中文關鍵字 守恆律方程組  自相似解  震波 
英文關鍵字 conservation law  self-similar solution  shock wave 
學科別分類
中文摘要 這篇論文中我們將要探討球面活塞問題。我們考慮一個放置在靜止氣體中的實心球面活塞,當它以等速擴張,推動前方沒有受到擾動的氣體時產生震波。我們的目標是在守恆律方程組自相似解的假設之下,藉由數值計算找出震波馬赫數與活塞無量綱速度之間的關係。此外,我們也藉由回顧兩篇文獻,探討由球面活塞運動所造成的會聚震波問題,與活塞在靜止氣體中運動所產生的震波級數解問題。
英文摘要 In this thesis, we consider a solid spherical piston in a quiescent polytropic gas, expanding at a constant speed, pushing out the undisturbed gas ahead of it, and then causing shock waves. Under the assumption of self-similar solutions, we discuss the relationship between the shock Mach number and the nondimensional velocity of the piston using numerical computations. In addition, we review literature related to converging shock waves caused by spherical piston motions and the corresponding series solutions of the piston motion for polytropic gas.
論文目次 1 Introduction 4

2 Preliminary 6
2.1 The Hyperbolic Conservation Laws 6
2.1.1 Weak Solutions of Conservation Laws 6
2.1.2 Elementary Waves 7
2.1.3 The Riemann Problem 12
2.2 Euler Equations 13
2.2.1 Conservation of Mass 14
2.2.2 Conservation of Momentum 15
2.2.3 Conservation of Energy 16

3 The Spherical Piston Problem 18
3.1 Euler Equations with Spherical Symmetry 19
3.2 Initial-Value Problem from Self-Similarity Assumption 20
3.3 Shock Mach Number and Nondimensional Velocity 23
3.4 Concluding Remarks 28

4 Converging Shock Waves of Spherical Piston Motions 31
4.1 Infinite Series Form of Locus for Converging Shock Waves 31
4.2 Finite Series Form of Shock Trajectory in a Polytropic Gas 36
4.3 Conclusion 41

References 42

Appendix A MATLAB Computations 43
A.1 Symbols and Equations 43
A.2 Numerical Method 45
A.3 Results and Discussions 47
參考文獻 [1] Courant, R. and Friedrichs, K. O., Supersonic Flow and Shock Waves, Springer-Verlag, New York, 1976.

[2] Evans, Lawrence C., Partial Differential Equations, American Mathematical Society, Providence, R.I., 1998.

[3] Kozmanov, M. Iu., On the motion of piston in a polytropic gas, J. Appl. Math. Mech., 41, 1152-1156, 1977.

[4] Pinchover, Yehuda and Rubinstein, Jacob, An Introduction to Partial Differential Equations, Cambridge University Press, New York, 2005.

[5] Rogers, M. H., Similarity flows behind strong shock waves, Quart. J. Mech. Appl. Math., 11, 411-422, 1958.

[6] Sachdev, P. L., Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems, Chapman & Hall/CRC, Boca Raton, 2000.

[7] Sachdev, P. L., Shock Waves and Explosions, Chapman & Hall/CRC, Boca Raton, 2004.

[8] Smoller, J., Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994.

[9] Strauss, W. A., Partial Differential Equations: An Introduction, Wiley, New York, 1992.

[10] Taylor, G. I., The air wave surrounding an expanding sphere, Proc. Roy. Soc. A, 186, 273-292, 1946.

[11] Van Dyke, M. and Guttmann, A. J., The converging shock wave from a spherical or cylindrical piston, J. Fluid Mech., 120, 451-462, 1982.

[12] Whitham, G. B., Linear and Nonlinear Waves, Wiley, New York, 1974.
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