偏心圆环空间内自然对流的确定性及混沌

doi:10.19596/j.cnki.1001-246x.8840

摘要/Abstract

摘要：

使用格子Boltzmann方法(LBM)研究偏心圆环空间内自然对流的非线性特性。通过最大李雅普诺夫指数图谱与游程检验的方法, 数学判定系统在高瑞利数时发展为混沌态; 根据数值解的相空间与功率谱密度(PSD)特性刻画系统通向混沌的历程。结果表明: 随着瑞利数(Ra)的增大, 偏心圆环系统的确定性解经过Hopf分岔转变为周期震荡解, 相空间轨迹由不动点转变为极限环; 随着Ra的进一步增加, 稳定的极限环分岔为二维环面, 系统进入拟周期态; 当Ra达到临界值时, 系统的相空间轨迹呈快速的指数级分离, 变得极其复杂, 其功率谱密度出现大量不可通约的基频, 混沌吸引子出现, Hopf分岔又一次发生, 最终进入混沌。

关键词: 偏心圆环, 自然对流, 格子玻尔兹曼方法, 分岔, 混沌

Abstract:

The nonlinear properties of natural convection in the space of an eccentric annulus are investigated using the lattice Boltzmann method (LBM). Firstly, the system is mathematically determined to develop into a chaotic state at high Rayleigh numbers through the maximum Lyapunov exponent spectrum and run test. Then, the process of the system transitioning to chaos is characterized based on the characteristics of numerical solution phase diagram and power spectral density (PSD). The results show that with the increase of Rayleigh number Ra, the solution of the eccentric annular system changes from deterministic steady-state solution to periodic oscillation solution through Hopf bifurcation, and the phase diagram trajectory changes from fixed point to limit cycle. With further increase of Rayleigh number, the stable limit cycle bifurcates into a two-dimensional torus, and the system enters a quasi-periodic state. When Rayleigh number Ra reaches a critical value, the phase diagram trajectory of the system exhibits rapid exponential separation, becomes extremely complex, and many incommensurable fundamental frequencies appear in its power spectral density. Chaotic attractors emerge, Hopf bifurcation occurs again, and eventually chaos is reached.

Key words: eccentric annulus, natural convection, lattice Boltzmann method, bifurcation, chaos

许一帆, 张虎明, 赵明. 偏心圆环空间内自然对流的确定性及混沌[J]. 计算物理, 2025, 42(1): 28-37.

Yifan XU, Huming ZHANG, Ming ZHAO. Determinism and Chaos of Natural Convection in Eccentric Annulus[J]. Chinese Journal of Computational Physics, 2025, 42(1): 28-37.

图/表 12

图1 几何模型

Fig.1 Physical model

图2 曲边边界示意图

Fig.2 Schematic diagram of curved edges model

图3 等温线的分布(a)文[4]; (b)本文

Fig.3 Distribution of isotherms (a) Ref.[4]; (b) this work

图4 径向无量纲温度分布(Ra=5×104，ε=(0, 180°))

Fig.4 Temperature distributions of radial dimensionless (Ra=5×104, ε=(0, 180°))

图5 不同Ra时系统的最大李雅普诺夫指数图谱( ε =(0, 180°))

Fig.5 The maximum Lyapunov exponent of the system for different Ra ( ε =(0, 180°))

表1 游程检验结果Ra=106，ε= (0, 180°)

Table 1 The results of run test at Ra=106 and ε= (0, 180°)

总个案数	临界值	游程数	Z值	P值
2 001	众数	353	-25.78	0
2 001	中位数	425	-27.80	0
2 001	平均数	356	-25.67	0

图6 监测点状态(a)无量纲温度；(b)u-v相图

Fig.6 State of monitoring point (a) dimensionless temperature; (b))u-v phase diagram

图7 ε =(-0.625, 180°) 时监测点状态(a)无量纲温度；(b))u-v相图；(c)功率谱密度

Fig.7 State of monitoring point at Ra=4×105, ε =(-0.625, 180°) (a) dimensionless temperature; (b))u-v phase diagram; (c) power spectral density

图8 Ra = 5 × 105，ε = (0, 180°) 时监测点状态(a)无量纲温度; (b)u-v相图; (c)功率谱密度

Fig.8 State of monitoring point at Ra = 5 × 105, ε = (0, 180°) (a) dimensionless temperature; (b) u-v phase diagram; (c) power spectral density

图9 Ra = 7 × 105，ε = (0, 180°) 时监测点状态(a)无量纲温度; (b)u-v相图; (c)功率谱密度

Fig.9 State of monitoring point at Ra = 7 × 105, ε = (0, 180°) (a) dimensionless temperature; (b) u-v phase diagram; (c) power spectral density

图10 Ra = 8.5 × 105，ε = (0, 180°) 时监测点状态(a)无量纲温度; (b)u-v相图; (c)功率谱密度

Fig.10 State of monitoring point at Ra = 8.5 × 105, ε = (0, 180°) (a) dimensionless temperature; (b) u-v phase diagram; (c) power spectral density

图11 Ra = 106，ε = (0, 180°) 时监测点状态(a)无量纲温度; (b)u-v相图; (c)功率谱密度

Fig.11 State of monitoring point at Ra = 106, ε = (0, 180°) (a) dimensionless temperature; (b) u-v phase diagram; (c) power spectral density

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