Institut für Mathematik und ihre Didaktik, Leuphana Universität Lüneburg, D-21335 Lüneburg, Germany.
Institut für Mathematik, Freie Universität Berlin, D-14195 Berlin, Germany.
Phys Rev E. 2019 Nov;100(5-1):053103. doi: 10.1103/PhysRevE.100.053103.
Coherent circulation rolls and their relevance for the turbulent heat transfer in a two-dimensional Rayleigh-Bénard convection model are analyzed. The flow is in a closed cell of aspect ratio four at a Rayleigh number Ra=10^{6} and at a Prandtl number Pr=10. Three different Lagrangian analysis techniques based on graph Laplacians (distance spectral trajectory clustering, time-averaged diffusion maps, and finite-element based dynamic Laplacian discretization) are used to monitor the turbulent fields along trajectories of massless Lagrangian particles in the evolving turbulent convection flow. The three methods are compared to each other and the obtained coherent sets are related to results from an analysis in the Eulerian frame of reference. We show that the results of these methods agree with each other and that Lagrangian and Eulerian coherent sets form basically a disjoint union of the flow domain. Additionally, a windowed time averaging of variable interval length is performed to study the degree of coherence as a function of this additional coarse graining which removes small-scale fluctuations that cause trajectories to disperse quickly. Finally, the coherent set framework is extended to study heat transport.
分析了二维瑞利-贝纳德对流模型中相干环流卷及其与湍流热传递的相关性。在瑞利数 Ra=10^6 和普朗特数 Pr=10 的情况下，流动处于纵横比为 4 的封闭单元中。使用三种基于图拉普拉斯的不同拉格朗日分析技术（距离谱轨迹聚类、时均扩散映射和基于有限元的动态拉普拉斯离散化）来监测无质量拉格朗日粒子在演化的湍流中的轨迹上的湍流场。将这三种方法进行了相互比较，并将得到的相干集与基于欧拉参考系的分析结果相关联。我们表明，这些方法的结果相互一致，并且拉格朗日和欧拉相干集基本上形成了流场的不相交并集。此外，还进行了带有可变时间间隔的窗口时间平均化，以研究相干性作为这种额外的粗粒化的函数，这种粗粒化去除了导致轨迹快速扩散的小尺度波动。最后，将相干集框架扩展到传热研究中。
