Couette-Taylor流的力学机理与能量转换  

Dynamical Mechanism and Energy Conversion of Couette-Taylor Flow

作　　者：王贺元 Wang Heyuan(College of Mathematics and Systematics Sciences,Shenyang Normal University,Shenyang 110034;College of Sciences,Liaoning University of Technology,Liaoning Jin'zhou 121001)

机构地区：[1]沈阳师范大学数学与系统科学学院,沈阳110034 [2]辽宁工业大学理学院,辽宁锦州121001

出　　处：《数学物理学报（A辑）》2020年第1期243-256,共14页Acta Mathematica Scientia

基　　金：国家自然科学基金(11572146);沈阳师范大学博士启动基金(054-91900302009)。

摘　　要：同轴圆筒间Couette-Taylor流问题是典型的旋转流动问题,它是层流到湍流过渡的范例,国内外众多学者对其进行了深入的研究.该文探讨Couette-Taylor流问题的力学机理与能量转换,通过将Couette-Taylor流三模混沌系统转换成Kolmogorov形系统,把系统的力矩分为四种类型:惯性力矩,内力矩,耗散力矩和外力矩.通过不同力矩的结合分析和研究了Couette-Taylor流产生混沌的关键因素和物理意义.研究了哈密顿能量,动能和势能之间的相互转换.讨论了能量与雷诺数之间的关系.研究表明四种力矩的耦合是产生混沌的必要条件,而且只有耗散力矩和驱动力矩(外力矩)相匹配时,系统才能产生混沌,其中任何三种力矩耦合均不可能产生混沌.圆筒旋转产生的外力矩供给系统能量,能量增长导致流动失稳,从而产生泰勒漩涡和混沌,进而得出了Couette-Taylor流的能量转换和物理意义.引进Casimir函数分析系统的动力学行为和能量转换,并估计混沌吸引子的界.Casimir函数反映了能量转换和轨道与平衡点间的距离,数值结果仿真出它们之间的关系.There have been a lot of investigations which concern with rotating flow between two concentric cylinders(abbreviate frequently as Couette-Taylor Flow),Couette-Taylor flow is the typical rotation flow problems,It provides a paradigm from laminar to turbulent transition.Dynamical mechanism and energy conversion of Couette-Taylor flow are investigated in this paper,the Couette-Taylor flow chaotic system is transformed into Kolmogorov type system,which is decomposed into four types of torques:inertial torque,internal torque,dissipation and external torque.By the combinations of different torques,key factors of chaos generation and the physical meaning of Couette-Taylor Flow are studied.The conversion among Hamiltonian energy,kinetic energy and potential energy is investigated.The relationship between the energies and the Reynolds number is discussed.It concludes that the combination of the four torques is necessary conditions to produce chaos,and only when the dissipative torques matches the driving(external)torques can the system produce chaos,any combination of three types of torques cannot produce chaos.The external torque,which is provided by the rotation of the cylinder,supply the energy of the system,and that leads to Taylor vortex and chaos,the physical meaning and energy conversion of Couette-Taylor flow system are investigated.The Casimir function is introduced to analyze the system dynamics,and its derivation is chosen to formulate energy conversion.The bound of chaotic attractor is obtained by the Casimir function and Lagrange multiplier.The Casimir function reflects the energy conversion and the distance between the orbit and the equilibria.These relationships are illustrated by numerical simulations.

关 键 词：Couette-Taylor流 力学机理 KOLMOGOROV系统 混沌 

分 类 号：O175.1[理学—数学]