平面不可压缩Navier-Stokes方程五模系统的力学机理及能量演化

Abstract

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

CLC Number:

O175.1

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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References 24

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$r$ -值范围	$0 < r < r_1$ 0 $r_1=5\sqrt{3/2}$	$r>r_1$ $r_e=28.71\quad r_g=28.72\quad r_h=28.73$
定点$O$	稳定结点	鞍结点(一个方向不稳,另两个方向稳定)
定点$P^{+}$和$P^{-}$	不存在	稳定结点	稳定焦点	稳定焦点	鞍点
系统(2.1)相空间中的运动情况	趋于稳定定态$O$	趋于稳定定态$P^{+}$或$P^{-}$	运动最终按螺旋线趋于$P^{+}$或$P^{-}$	同左,但越靠近$r_h$,轨线在$P^{+}$和$P^{-}$之间来回跳动的越频繁,出现暂态混沌,最终趋于$P^{+}$或$P^{-}$	不稳定极限环(亚临界霍普夫分岔)
动能	最小值	逐渐增大		保持增大	增长
Casimir函数	最小值	逐渐增大		保持增大	增长
$D_1$与$D_2$之和	不存在	逐渐递增		递增增大	逐渐

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

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