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系統識別號 U0026-0812200914083589
論文名稱(中文) 圓柱震波的應用
論文名稱(英文) Application of Cylindrical Shock Waves
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) 鄭夙君
研究生(英文) Su-chun Cheng
學號 l1694105
學位類別 碩士
語文別 英文
論文頁數 45頁
口試委員 口試委員-陳旻宏
口試委員-侯世章
指導教授-連文璟
中文關鍵字 起始值問題  常微分方程系統  自我相似  震波  流星軌跡  活塞 
英文關鍵字 initial-value problem  system of conservation laws  self-similar  Rankine-Hugoniot condition  shock wave  piston  system of ordinary differential equations  meteor 
學科別分類
中文摘要 在這篇論文中, 我們討論的是圓柱活塞問題。在柱對稱系統中, 當在靜止均勻氣體中的圓柱活塞, 均勻地向外擴張時, 在外圍會產生圓柱震波。由解的自我相似性的假設, 此活塞問題能夠被簡化為起始條件在震波上的常微分方程系統的起始值問題。另外, 我們考慮活塞周圍的氣體速度等於活塞速度的邊界條件。代入不同的震波馬赫數, 我們可以由對常微分方程系統的數值積分得到活塞位置以及活塞無因次速度。我們的目標是找出震波馬赫數與活塞無因次速度的關係。除此之外, 我們也整理了S. C. Lin[6]所討論的在流星軌跡周圍所產生的圓柱震波的論文。
英文摘要 In this thesis, we discuss a cylindrical piston problem. In the case of cylindrical symmetry, as a cylindrical piston uniformly expands and pushes out undisturbed uniform polytropic gas ahead of itself, a cylindrical shock wave occurs outside the piston. Resulting from the assumption of self-similarity of the flow, this piston problem is simplified as the initial-value problem of a system of ordinary differential equations with the initial condition at the shock locus. The kinematic condition is considered as the boundary condition at the piston. For a different shock Mach number, we can obtain the position of the inner surface of a piston and the nondimensional velocity of a piston by the numerical integration of the system of ordinary differential equations. Our goal is to find the relationship between the shock Mach number and the nondimensional velocity of a piston. Additionally, we summarize S. C. Lin’s paper[6] regarding the cylindrical shock waves which occur around the luminous trails of meteors.
論文目次 1 Introduction 5
2 Preliminary 7
2.1 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . 7
2.1.2 Conservation of Momentum . . . . . . . . . . . . . . . 8
2.1.3 Conservation of Energy . . . . . . . . . . . . . . . . . . 9
2.2 Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Shock Waves, RarefactionWaves, and Contact Discontinuities
. . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.3 Riemann’s Problem . . . . . . . . . . . . . . . . . . . . 17
3 Self-similar Solutions in a Cylindrical System 19
3.1 A Uniformly Expanding Shock Wave . . . . . . . . . . . . . . 21
3.2 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . 24
4 Application 31
4.1 An Intense Cylindrical Shock Wave Problem . . . . . . . . . . 31
4.2 The Relationship between the Shock Radius and the Distance behind a Meteoroid . . . . . . . . . . . . . . . . . . . . . . . . 34
References 35
Appendix A Numerical Data 37
A.1 Computation Process . . . . . . . . . . . . . . . . . . . . . . . 37
A.2 The Relationship between the Shock Mach Number and the
Nondimensional Velocity of the Piston . . . . . . . . . . . . . 42
參考文獻 [1] Courant, R. and Friedrichs, K. O., Supersonic Flow and Shock Waves, Springer-Verlag, New York, 1976.
[2] Evans, L. C., Partial Differential Equations, American Mathematical Society, Providence, R.I., 1998.
[3] John, James E. A., Gas Dynamics, Allyn and Bacon, Boston, 1984.
[4] LaPaz, Lincoln and LaPaz, Jean, Space Nomads: Meteorites in Sky, Field and Laboratory, Holiday House, New York, 1961.
[5] LeVeque, R. J., Numerical Methods for Conservation Laws, Birkh¨auser Verlag, Basel, 1992.
[6] Lin, S. C., Cylindrical Shock Waves Produced by Instantaneous Energy Release, J. Appl. Phys., 25, 54–57, 1954.
[7] Norton, O. R., The Cambridge Encyclopedia of Meteorites, Cambridge University Press, Cambridge, 2002.
[8] Pinchover, Yehuda and Rubinstein, Jacob, An Introduction to Partial Differential Equations, Cambridge University Press, Cambridge, 2005.
[9] Rogers, M. H., Similarity Flows behind Strong Shock Waves, Quart. J. Mech. Appl. Math., 11, 411–422, 1958.
[10] Sachdev, P. L., Shock Waves and Explosions, Chapman & Hall/CRC, Boca Raton, 2004.
[11] Serre, Denis, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves, Cambridge University Press, Cambridge, 1999.
[12] Smoller, Joel, ShockWaves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994.
[13] Strauss, W. A., Partial Differential Equations: An Introduction, Wiley, New York, 1992.
[14] Taylor, G. I., The Air Wave Surrounding an Expanding Sphere, Pro. Roy. Soc. A, 186, 273–292, 1946.
[15] Whitham, G. B., Linear and Nonlinear Waves, Wiley, New York, 1999.
[16] International Meteor Organization, http://www.imo.net/
[17] Digital Images of the Sky, http://www.allthesky.com/
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