Abstract
Convection driven by a spatially non-uniform internal heat source between two horizontal isothermal walls is studied by theoretical analysis and numerical simulation, in order to explore the bounds of the temperature and the vertical heat flux. Specifically, the rigorous lower bound of the weighted average temperature ⟨QT⟩ is derived analytically, by decomposing the temperature field into a background profile and a fluctuation part. This bound obtained for the first time to consider non-uniform heat sources is found to be compatible with the existing bound obtained in uniform internal heat convection. Of physical importance, an analytical relationship is derived as an inequality connecting ⟨QT⟩ and the average vertical heat flux ⟨wT⟩, by employing the average heat flux on the bottom wall (qb) as an intermediary variable. It clarifies the intrinsic relation between the lower bound of ⟨QT⟩ and the upper bound of ⟨wT⟩, namely, these two bounds are essentially equivalent providing an easy way to obtain one from another. Furthermore, the analytical bounds are extensively demonstrated through a comprehensive series of direct numerical simulations.
摘要
通过理论分析和数值模拟研究了两个水平等温壁面之间受非均匀内部热源驱动的对流, 以探索温度和数值热通量的界. 具体 而言, 通过将温度分解为定常的背景温度剖面和脉动温度两部分, 解析地推导出了加权平均温度⟨QT⟩的严格下界. 作为首个考虑非均 匀内部热源的解析界, 它能与已有的用于均匀内部加热对流的界兼容. 更具物理意义的是, 通过使用下壁面平均热通量(qb)作为中间变 量, 我们得到了一个解析关系用于构建⟨QT⟩和平均竖直热通量⟨wT⟩的不等关系. 它揭示了⟨QT⟩的下界和⟨wT⟩的上界之间的内在关联, 即这两个界本质上是等价的, 它提供了一种简单的方法将一个界转化为另一个. 此外,我们通过一系列全面的数值模拟,对解析界进 行了广泛的测试验证.
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Acknowledgements
The work was supported by the National Natural Science Foundation of China (Grant Nos. 92252202, 92152301, 12293000, 12293002, 12302320, and 12388101), and the Fundamental Research Funds for the Central Universities. The authors thank X.-Y. Lu for many useful and illuminating discussions.
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Author contributions Liangbing Chen provided research concepts and overarching research goals, developed the methodology and conducted the research and investigation process. Liangbing Chen completed early work on theoretical derivation and numerical simulations. An-Kang Gao and Zimo Liao helped to conduct the investigation process and expanded the theoretical derivation. Zhenhua Wan and Zimo Liao repeatedly checked and validated the theoretical derivation process and numerical simulations. Liangbing Chen presented the published work and visualized the simulation data. Liangbing Chen and An-Kang Gao wrote the first draft of the manuscript. An-Kang Gao and Nansheng Liu revised and edited the final version, and supervised the advancement of the research. Nansheng Liu provided the computing resources and financial support for the research.
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Chen, L., Gao, AK., Liao, Z. et al. Temperature and heat flux bounds of convection driven by non-uniform internal heating. Acta Mech. Sin. 40, 323630 (2024). https://doi.org/10.1007/s10409-024-23630-x
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DOI: https://doi.org/10.1007/s10409-024-23630-x