Benard对流问题五模类Lorenz方程组混沌行为的数值模拟

Numerical simulation of the chaos behavior of five-dimensional Lorenz-like system of BENDARD convection problem

考虑底部加热的两平行板间流层的Benard对流问题,其实际背景源于简化的大气对流模型,该问题可以用Navier-Stokes方程与热传导方程的耦合方程来描述,这是无穷维动力系统,其动力学行为分析是极具挑战性的。为探讨其对流现象采用有限模态分析方法,对其速度场和温度场等相关变量进行二维傅立叶展开,经分析及复杂运算得到新五模类Lorenz方程组,对此方程组的动力学行为进行了数值模拟。基于分岔图、最大李雅普诺夫指数、庞加莱截面、功率谱和返回映射展现了系统混沌行为的普适特征。数值结果表明简化的Benard对流问题的新五模类Lorenz系统存在混沌现象,这个简化的模型从侧面描述了简化的大气对流问题的部分特征。

In this paper we study the Benard convection problem which is described by Navier-Stokes equation and heat conduction equation. This is an infinite dimensional dynamical system, and the analysis of its dynamic behavior is extremely challenging. Finite modal analysis is used to investigate the convection phenomena. For the fluid velocity field and the temperature field, using Fourier series we obtain several five-modes Lorenz equations. The chaos behaviors are simulated numerically by computer according to the changing of Rayleigh number. Based on numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum, Poincare section, power spectrum and return map of the system, some chaos behavior of the system are revealed.The numerical results show that the new five-mode Lorenz system of the simplified Benard convection problem is chaotic, and this simplified model describes some characteristics of the simplified atmospheric convection problem in a different way.