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Three-dimensional bifurcations in a cubic cavity due to buoyancy-driven natural convection

机译：浮力驱动的自然对流导致立方腔中的三维分叉

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摘要

Rich and complex buoyancy-driven flow field due to natural convection will be studied numerically over a wide range of Rayleigh numbers in a cubic cavity by virtue of the simulated bifurcation diagram, limit cycle, power spectrum and phase portrait. When increasing the Rayleigh number, the predicted flow is found to evolve from the conductive state to the state with the onset of convection, which is featured with the steady and symmetric laminar solution, and then to the asymmetric state (pitchfork bifurcation), which will not be discussed in this paper. As the Rayleigh number was further increased, a limit cycle branching from the fixed point of the investigated dynamical system is observed. Supercritical Hopf bifurcation is confirmed to be the birth of the orbitally stable limit cycle that separates the vortex flow into an inner unstable region (moving away from the vortex coreline) and an outer stable region (moving towards the vortex coreline). As the Rayleigh number is increased still, the investigated buoyancy-driven flow became increasingly destabilized through quasi-periodic bifurcation and then through two predicted frequency-doubling bifurcations. Thanks to the power spectrum analysis, bifurcation scenario was confirmed to have an initially single harmonic frequency, which is featured with a driving amplitude. Then an additional ultraharmonic frequency showed its presence. Prior to chaos, in the five predicted arithmetically related frequencies there exists one frequency that is incommensurate to the other two fundamental frequencies. This computational study enlightens that the investigated nonlinear system, which involves frequency-doubling bifurcations, loses its stability to a quasi-periodic bifurcation featured with the formation of a subharmonic frequency. Subsequent to the formation of three frequency-doubling bifurcations and one quasi-periodic bifurcation, an infinite number of frequencies was observed in flow conditions with the continuously increasing Rayleigh numbers. Finally, the chaotic attractor was predicted to be evolved from the strange attractor in the corresponding phase portraits.
机译：将通过模拟分叉图，极限环，功率谱和相图，对立方空腔内宽范围的瑞利数在数值上研究由于自然对流而产生的丰富而复杂的浮力驱动流场。当增加瑞利数时，发现预测的流量从导电状态演变为具有对流且以稳定和对称层流为特征的对流，然后发展为非对称状态（干草叉分叉），本文不讨论。随着瑞利数的进一步增加，观察到从研究的动力学系统的固定点分支出来的极限循环。超临界霍夫夫分叉被证实是轨道稳定极限环的产生，该极限稳定环将涡流分为内部不稳定区域（远离涡旋中心线）和外部稳定区域（朝向涡旋中心线）。随着瑞利数的增加，通过准周期分叉，然后通过两个预测的倍频分叉，浮力驱动的流量变得越来越不稳定。由于进行了功率谱分析，因此确定了分叉方案最初具有单个谐波频率，该频率具有驱动幅度。然后，另一个超谐波频率显示出它的存在。在混乱之前，在五个预测的算术相关频率中，存在一个与其他两个基本频率不相称的频率。该计算研究表明，所研究的涉及倍频分叉的非线性系统失去了对以亚谐波频率形成为特征的准周期分叉的稳定性。在形成三个倍频分叉和一个准周期分叉之后，随着瑞利数的不断增加，在流动条件下观察到了无限多个频率。最终，混沌吸引子被预测为在相应的相画像中从奇怪吸引子演化而来。

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