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研究生：	林冠廷 Lin, Kuan-Ting
論文名稱：	球面活塞問題中自相似解之探討 The Self-Similar Flow of the Spherical Piston Problem
指導教授：	連文璟 Lien, Wen-Ching
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系應用數學碩博士班 Department of Mathematics
論文出版年：	2008
畢業學年度：	96
語文別：	英文
論文頁數：	50
中文關鍵詞：	守恆律方程組、自相似解、震波
外文關鍵詞：	conservation law, self-similar solution, shock wave
相關次數：	點閱：69 下載：5
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查詢本校圖書館目錄查詢臺灣博碩士論文知識加值系統勘誤回報

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。

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.

Introduction  4

Preliminary  6
1 The Hyperbolic Conservation Laws  6
1.1 Weak Solutions of Conservation Laws  6
1.2 Elementary Waves  7
1.3 The Riemann Problem  12
2 Euler Equations  13
2.1 Conservation of Mass  14
2.2 Conservation of Momentum  15
2.3 Conservation of Energy  16

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

Converging Shock Waves of Spherical Piston Motions  31
1 Infinite Series Form of Locus for Converging Shock Waves  31
2 Finite Series Form of Shock Trajectory in a Polytropic Gas  36
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
                                    

[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.

校內：立即公開
校外：2008-04-30公開

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