Determinism and Chaos of Natural Convection in Eccentric Annulus

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

Abstract

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

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.

Figures/Tables 12

Fig.1 Physical model

Fig.2 Schematic diagram of curved edges model

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

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

Fig.5 The maximum Lyapunov exponent of the system for different Ra ( 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

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

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

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

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

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

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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