平面不可压缩Navier-Stokes方程五模系统的力学机理及能量演化

数学物理学报 ›› 2020, Vol. 40 ›› Issue (2): 315-327.

平面不可压缩Navier-Stokes方程五模系统的力学机理及能量演化

王贺元^1,²()

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

收稿日期:2018-04-11 出版日期:2020-04-26 发布日期:2020-05-21
作者简介:王贺元, E-mail:987236994@qq.com
基金资助:
国家自然科学基金(11572146);沈阳师范大学博士启动基金(054-91900302009)

Dynamical Mechanism and Energy Evolution of a Five-Modes System of the Navier-Stokes Equations For a Two-Dimensional Incompressible Fluid on a Torus

Heyuan Wang^1,²()

¹ College of Mathematics and Systematics Sciences, Shenyang Normal University, Shenyang 110034
² College of Sciences, Liaoning University of Technology, Liaoning Jin'zhou 121001

Received:2018-04-11 Online:2020-04-26 Published:2020-05-21
Supported by:
the NSFC(11572146);the Doctor Science Foundation of Shenyang Normal University(054-91900302009)

摘要/Abstract

摘要：

该文研究了平面不可压缩Navier-Stokes方程五模系统的力学机理及能量演化问题，通过将五模混沌系统转换成Kolmogorov形系统，把系统的力矩分为三种类型：惯性力矩，耗散力矩和外力矩.通过不同力矩的结合分析和研究了系统产生混沌的关键因素和物理意义.讨论了能量与雷诺数之间的关系.研究表明三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，其中任何两种力矩耦合均不可能产生混沌.外力矩给系统提供能量，导致系统失稳出现分岔与混沌.引进Casimir函数分析系统的动力学行为和能量演化，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.

关键词: 力学机理, 能量演化, Kolmogorov系统, 混沌

Abstract:

In this paper we study dynamical mechanism and energy evolution of a five-modes system of the Navier-Stokes equations for a two-dimensional incompressible fluid on a torus. The five-modes system is transformed into Kolmogorov type system, which is decomposed into three types of torques:inertial torque, dissipation and external torque. Combining different torques, key factors of chaos generation and the physical meaning of the five-modes system are studied. The evolution of energy is investigated, the relationship between the energies and the Reynolds number is discussed. We conclude that the combination of the three torques is necessary conditions to produce chaos, and only when the dissipative torques match the driving torques (external torque) the system can produce chaos. While any combination of two types of torques cannot produce chaos. The external torque supply the energy for the system, and that leads to bifurcation and chaos. The Casimir function is introduced to analyze the system dynamics, and its derivation is chosen to formulate energy evolution. The bound of chaotic attractor is obtained by the Casimir function and Lagrange multiplier. We find that the Casimir function reflects the energy evolution and the distance between the orbit and the equilibria.

Key words: Dynamical Mechanism, Kolmogorov system, Chaos

中图分类号:

O175.1

王贺元. 平面不可压缩Navier-Stokes方程五模系统的力学机理及能量演化[J]. 数学物理学报, 2020, 40(2): 315-327.

Heyuan Wang. Dynamical Mechanism and Energy Evolution of a Five-Modes System of the Navier-Stokes Equations For a Two-Dimensional Incompressible Fluid on a Torus[J]. Acta mathematica scientia,Series A, 2020, 40(2): 315-327.

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