SpringerLink
Log in

Temperature and heat flux bounds of convection driven by non-uniform internal heating

非均匀内部热源驱动对流的温度和热通量界

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

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**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**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**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**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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Y. Du, M. Zhang, and Y. Yang, Thermal convection driven by a heat-releasing scalar component, Acta Mech. Sin. 38, 321584 (2022).

    Article  Google Scholar 

  2. Y. Yang, Double diffusive convection in the finger regime for different Prandtl and Schmidt numbers, Acta Mech. Sin. 36, 797 (2020).

    Article  MathSciNet  Google Scholar 

  3. G. Schubert, D. L. Turcotte, and P. Olson, Mantle Convection in the Earth and Planets (Cambridge University Press, Cambridge, 2001).

    Book  Google Scholar 

  4. R. Kippenhahn, A. Weigert, and A. Weiss, Stellar Structure and Evolution (Springer, Berlin, Heidelberg, 1990).

    Book  Google Scholar 

  5. K. A. Emanuel, Atmospheric Convection (Oxford University Press, Oxford, 1994).

    Book  Google Scholar 

  6. Y. Shen, and O. Zikanov, Thermal convection in a liquid metal battery, Theor. Comput. Fluid Dyn. 30, 275 (2016), ar**v: 1507.08315.

    Article  Google Scholar 

  7. Z. Lu, G. Liu, and B. Wang, Flow structure and heat transfer of electrothermo-convection in a dielectric liquid layer, Phys. Fluids 31, 064103 (2019).

    Article  Google Scholar 

  8. D. J. Tritton, and M. N. Zarraga, Convection in horizontal layers with internal heat generation. Experiments, J. Fluid Mech. 30, 21 (1967).

    Article  Google Scholar 

  9. A. Limare, C. Jaupart, E. Kaminski, L. Fourel, and C. G. Farnetani, Convection in an internally heated stratified heterogeneous reservoir, J. Fluid Mech. 870, 67 (2019).

    Article  MathSciNet  Google Scholar 

  10. F. A. Kulacki, and R. J. Goldstein, Thermal convection in a horizontal fluid layer with uniform volumetric energy sources, J. Fluid Mech. 55, 271 (1972).

    Article  Google Scholar 

  11. V. Bouillaut, S. Lepot, S. Aumaître, and B. Gallet, Transition to the ultimate regime in a radiatively driven convection experiment, J. Fluid Mech. 861, R5 (2019), ar**v: 2010.10936.

    Article  MathSciNet  Google Scholar 

  12. Y. Tasaka, and Y. Takeda, Effects of heat source distribution on natural convection induced by internal heating, Int. J. Heat Mass Transfer 48, 1164 (2005).

    Article  Google Scholar 

  13. D. Goluskin, and E. P. van der Poel, Penetrative internally heated convection in two and three dimensions, J. Fluid Mech. 791, R6 (2016), ar**v: 1511.05966.

    Article  MathSciNet  Google Scholar 

  14. D. Goluskin, and E. A. Spiegel, Convection driven by internal heating, Phys. Lett. A 377, 83 (2012), ar**v: 1210.8154.

    Article  Google Scholar 

  15. Z. L. **a, C. B. Zhao, J. Z. Wu, B. F. Wang, and K. L. Chong, Temperature response to periodic modulation in internal heating convection, Phys. Fluids 34, 125133 (2022).

    Article  Google Scholar 

  16. C. Jain, and V. S. Solomatov, Onset of convection in internally heated fluids with strongly temperature-dependent viscosity, Phys. Fluids 34, 096604 (2022).

    Article  Google Scholar 

  17. G. Kefayati, Internally heated convection of viscoplastic fluids in enclosures using a lattice Boltzmann method, Phys. Fluids 35, 013108 (2023).

    Article  Google Scholar 

  18. Q. Wang, D. Lohse, and O. Shishkina, Scaling in internally heated convection: A unifying theory, Geophys. Res. Lett. 48, e2020GL091198 (2021), ar**v: 2010.05789.

    Article  Google Scholar 

  19. D. Goluskin, Internally Heated Convection and Rayleigh-Bénard Convection (Springer, Berlin, 2016).

    Book  Google Scholar 

  20. A. J. Harfash, Three dimensional simulations of penetrative convection in a porous medium with internal heat sources, Acta Mech. Sin. 30, 144 (2014).

    Article  MathSciNet  Google Scholar 

  21. A. Mahajan, and M. K. Sharma, Penetrative convection in magnetic nanofluids via internal heating, Phys. Fluids 29, 034101 (2017).

    Article  Google Scholar 

  22. X. He, and P. Tong, Space-time correlations in turbulent Rayleigh-Bénard convection, Acta Mech. Sin. 30, 457 (2014).

    Article  MathSciNet  Google Scholar 

  23. H. Y. Zou, W. F. Zhou, X. Chen, Y. Bao, J. Chen, and Z. S. She, Boundary layer structure in turbulent Rayleigh-Bénard convection in a slim box, Acta Mech. Sin. 35, 713 (2019), ar**v: 1901.07913.

    Article  MathSciNet  Google Scholar 

  24. X. Zhu, and Q. Zhou, Flow structures of turbulent Rayleigh-Bénard convection in annular cells with aspect ratio one and larger, Acta Mech. Sin. 37, 1291 (2021).

    Article  MathSciNet  Google Scholar 

  25. I. Tobasco, D. Goluskin, and C. R. Doering, Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems, Phys. Lett. A 382, 382 (2018), ar**v: 1705.07096.

    Article  MathSciNet  Google Scholar 

  26. G. Fantuzzi, A. Arslan, and A. Wynn, The background method: Theory and computations, Phil. Trans. R. Soc. A. 380, 20210038 (2022), ar**v: 2107.11206.

    Article  MathSciNet  Google Scholar 

  27. G. Fantuzzi, and A. Wynn, Optimal bounds with semidefinite programming: An application to stress-driven shear flows, Phys. Rev. E 93, 043308 (2016), ar**v: 1512.05615.

    Article  Google Scholar 

  28. C. R. Doering, F. Otto, and M. G. Reznikoff, Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh-Bénard convection, J. Fluid Mech. 560, 229 (2006).

    Article  MathSciNet  Google Scholar 

  29. S. Chernyshenko, Relationship between the methods of bounding time averages, Phil. Trans. R. Soc. A. 380, 20210044 (2022).

    Article  MathSciNet  Google Scholar 

  30. D. Goluskin, and G. Fantuzzi, Bounds on mean energy in the Kuramoto-Sivashinsky equation computed using semidefinite programming, Nonlinearity 32, 1705 (2019), ar**v: 1802.08240.

    Article  MathSciNet  Google Scholar 

  31. L. Lu, C. R. Doering, and F. H. Busse, Bounds on convection driven by internal heating, J. Math. Phys. 45, 2967 (2004).

    Article  MathSciNet  Google Scholar 

  32. A. Arslan, G. Fantuzzi, J. Craske, and A. Wynn, Bounds on heat transport for convection driven by internal heating, J. Fluid Mech. 919, A15 (2021), ar**v: 2102.06458.

    Article  MathSciNet  Google Scholar 

  33. A. Arslan, G. Fantuzzi, J. Craske, and A. Wynn, Rigorous scaling laws for internally heated convection at infinite Prandtl number, J. Math. Phys. 64, 023101 (2023), ar**v: 2205.03175.

    Article  MathSciNet  Google Scholar 

  34. A. Kumar, A. Arslan, G. Fantuzzi, J. Craske, and A. Wynn, Analytical bounds on the heat transport in internally heated convection, J. Fluid Mech. 938, A26 (2022), ar**v: 2110.10344.

    Article  MathSciNet  Google Scholar 

  35. J. P. Whitehead, and C. R. Doering, Internal heating driven convection at infinite Prandtl number, J. Math. Phys. 52, 093101 (2011), ar**v: 1104.2792.

    Article  MathSciNet  Google Scholar 

  36. C. R. Doering, and P. Constantin, Variational bounds on energy dissipation in incompressible flows. III. Convection, Phys. Rev. E 53, 5957 (1996).

    Article  Google Scholar 

  37. G. Fantuzzi, A. Arslan, and A. Wynn, The background method: Theory and computations, Phil. Trans. R. Soc. A. 380, 20210038 (2022), ar**v: 2107.11206.

    Article  MathSciNet  Google Scholar 

  38. Y. Zhang, Q. Zhou, and C. Sun, Statistics of kinetic and thermal energy dissipation rates in two-dimensional turbulent Rayleigh-Bénard convection, J. Fluid Mech. 814, 165 (2017).

    Article  MathSciNet  Google Scholar 

  39. Y. Wang, W. Xu, X. He, H. Yik, X. Wang, J. Schumacher, and P. Tong, Boundary layer fluctuations in turbulent Rayleigh-Bénard convection, J. Fluid Mech. 840, 408 (2018).

    Article  Google Scholar 

  40. A. Xu, L. Shi, and H. D. **, Statistics of temperature and thermal energy dissipation rate in low-Prandtl number turbulent thermal convection, Phys. Fluids 31, 125101 (2019), ar**v: 1911.04042.

    Article  Google Scholar 

  41. J. Höpken, and K. G. Mooney, The openfoam© technology primer, 2021.

  42. C. J. Greenshields, OpenFOAM v6 User Guide (OpenFOAM Foundation Ltd., London, 2015).

    Google Scholar 

  43. A. Xu, X. Chen, F. Wang, and H. D. **, Correlation of internal flow structure with heat transfer efficiency in turbulent Rayleigh-Bénard convection, Phys. Fluids 32, 105112 (2020), ar**v: 2009.07675.

    Article  Google Scholar 

  44. R. Rosa, and R. Temam, Optimal minimax bounds for time and ensemble averages of dissipative infinite-dimensional systems with applications to the incompressible navier-stokes equations, ar**v: 2010.06730.

Download references

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.

Author information

Authors and Affiliations

  1. Department of Modern Mechanics, University of Science and Technology of China, Hefei, 230026, China

    Liangbing Chen  (陈良兵), An-Kang Gao  (高安康), Zimo Liao  (廖紫默), Zhenhua Wan  (万振华) & Nansheng Liu  (刘难生)

Authors
  1. Liangbing Chen  (陈良兵)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. An-Kang Gao  (高安康)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zimo Liao  (廖紫默)
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhenhua Wan  (万振华)
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Nansheng Liu  (刘难生)
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

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.

Corresponding authors

Correspondence to An-Kang Gao  (高安康) or Nansheng Liu  (刘难生).

Ethics declarations

Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10409-024-23630-x

Search

Navigation