Natural convection over a roof-shaped triangular surface is investigated using direct numerical simulations. The Rayleigh number (Ra) was varied from 1 to $5\times {10}^6$ with air as working fluid (Prandtl number of 0.71) at a fixed geometrical aspect ratio of 0.1, defined as the ratio of roof height to half-width. The transition route from a steady flow to a chaotic flow on the surface is characterized by the topological method with the increase of Ra. A weak flow, dominated by conduction, occurs when Ra was relatively small. As Ra increases, the convective flow becomes stronger and a sequence of bifurcations is found. Between $\mathrm{Ra}={10}^2$ and ${10}^3$, a primary pitchfork bifurcation occurs. Secondary and tertiary pitchfork bifurcations are observed in the range $\mathrm{Ra}=\left[{10}^3,{10}^4\right]$ and $\left[{10}^4,{10}^5\right]$, respectively. After another pitchfork bifurcation at $\mathrm{Ra}=\left[1.4,1.5\right]\times {10}^6$, which makes the plume tilt to either side of the roof top edge, a Hopf bifurcation is observed in $\mathrm{Ra}=\left[1.9,2\right]\times {10}^6$, after which both the slope flow and plume become periodic. This is followed by further bifurcations including a period doubling bifurcation at $\mathrm{Ra}\approx 3\times {10}^6$ and a quasiperiodic bifurcation firstly arising at $\mathrm{Ra}\approx 3.4\times {10}^6$. Finally, the flow becomes chaotic for $\mathrm{Ra}>3.7\times {10}^6$. The state space, the maximum Lyapunov exponent, the fractal dimension, and the power spectral density are presented to analyze the flows in the transition to chaos. This work is a comprehensive description of the flow transition from steady state to chaos on surface of a section-triangular roof that is pertinent to various settings where fluid flow develops.

中文翻译：

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截面三角形屋顶上自然对流从稳定流向混沌流的过渡

使用直接数值模拟研究了屋顶形三角形表面上的自然对流。瑞利数（Ra）从1到$5\times {10}^6$以空气作为工作流体（普朗特数为0.71），固定几何长宽比为0.1，定义为顶高与半角之比。随着Ra的增加，采用拓扑学方法对表面上从稳态流向混沌流的过渡路径进行了表征。当Ra相对较小时，会发生以导电为主的微弱流动。随着Ra的增加，对流流动变得更强，并且发现了一系列分叉。之间$镭={10}^2$ 和 ${10}^3$，发生一次主干草叉分叉。在此范围内观察到二级和三级干草叉分叉$镭=\left[{10}^3，{10}^4\right]$ 和 $\left[{10}^4，{10}^5\right]$， 分别。在另一个干草叉分叉后$镭=\left[1.4，1.5\right]\times {10}^6$，这会使烟流向屋顶顶部边缘的任一侧倾斜，在 $镭=\left[1.9，2\right]\times {10}^6$之后，坡度流和羽流都变为周期性。随后是进一步的分叉，包括在$镭\approx 3\times {10}^6$ 准周期分叉首先出现在 $镭\approx 3.4\times {10}^6$。最终，流程变得混乱$镭>3.7\times {10}^6$。给出了状态空间，最大Lyapunov指数，分形维数和功率谱密度，以分析过渡到混沌过程中的流动。这项工作是对截面三角形屋顶表面从稳态到混沌的流动过渡的全面描述，该屋顶过渡与流体流动发展的各种设置有关。