Skip to main content
SpringerLink
Log in

Exact mathematical formulas for wall-heat flux in compressible turbulent channel flows

可压缩槽道湍流中壁面热流的精确数学公式

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

Abstract

In this paper, several exact expressions for the mean heat flux at the wall (qw) for the compressible turbulent channel flows are derived by using the internal energy equation or the total energy equation. Two different routes, including the FIK method and the RD method, can be applied. The direct numerical simulation data of compressible channel flows at different Reynolds and Mach numbers verify the correctness of the derived formulas. Discussions related to the different energy equations, and different routes are carried out, and we may arrive at the conclusion that most of the formulas derived in the present work are just mathematical ones and that they generally are lacking in clear physical interpretation in our opinion. They can be used to estimate qw, but might not be suitable for exploring the underlying physics.

摘要

本文从内能方程和总能量方程两种不同的能量方程出发, 采用FIK方法和RD方法两种不同的推导方法, 推导了几组关于可压缩槽道湍流的壁面平均热流(qw)的数学公式, 并用不同雷诺数和马赫数下的可压缩槽道湍流直接数值模拟数据验证了这些公式的正确性和准确性. 我们从能量方程和推导方法的选择上对这些公式展开了详细讨论, 并得到结论: 本文中推导出来的大多数公式仅仅只是准确的数学公式, 它们缺乏明确的物理含义. 它们可以用来估算q w, 但是可能并不适合用来探索其中的物理机理.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. A. Leyva, The relentless pursuit of hypersonic flight, Phys. Today 70, 30 (2017).

    Article  Google Scholar 

  2. J. Urzay, Supersonic Combustion in Air-Breathing Propulsion Systems for Hypersonic Flight, Annu. Rev. Fluid Mech. 50, 593 (2018).

    Article  MathSciNet  Google Scholar 

  3. G. V. Candler, Rate Effects in Hypersonic Flows, Annu. Rev. Fluid Mech. 51, 379 (2019).

    Article  MathSciNet  Google Scholar 

  4. P. Bradshaw, Compressible Turbulent Shear Layers, Annu. Rev. Fluid Mech. 9, 33 (1977).

    Article  Google Scholar 

  5. S. K. Lele, Compressibility effects on turbulence, Annu. Rev. Fluid Mech. 26, 211 (1994).

    Article  MathSciNet  Google Scholar 

  6. R. Friedrich, Compressible turbulent flows: Aspects of prediction and analysis, Z. Angew. Math. Mech. 87, 189 (2007).

    Article  Google Scholar 

  7. T. B. Gatski, and J.-P. Bonnet, Compressibility, Turbulence and High Speed Flow (Academic Press, New York, 2013).

    Google Scholar 

  8. Q. Wang, J. P. Li, W. Zhao, and Z. L. Jiang, Comparative study on aerodynamic heating under perfect and nonequilibrium hypersonic flows, Sci. China-Phys. Mech. Astron. 59, 624701 (2016).

    Article  Google Scholar 

  9. K. Fukagata, K. Iwamoto, and N. Kasagi, Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows, Phys. Fluids 14, L73 (2002).

    Article  Google Scholar 

  10. T. Gomez, V. Flutet, and P. Sagaut, Contribution of Reynolds stress distribution to the skin friction in compressible turbulent channel flows, Phys. Rev. E 79, 035301 (2009).

    Article  Google Scholar 

  11. N. Renard, and S. Deck, A theoretical decomposition of mean skin friction generation into physical phenomena across the boundary layer, J. Fluid Mech. 790, 339 (2016).

    Article  MathSciNet  Google Scholar 

  12. M. Yoon, J. Ahn, J. Hwang, and H. J. Sung, Contribution of velocity-vorticity correlations to the frictional drag in wall-bounded turbulent flows, Phys. Fluids 28, 081702 (2016).

    Article  Google Scholar 

  13. F. Mehdi, and C. M. White, Integral form of the skin friction coefficient suitable for experimental data, Exp. Fluids 50, 43 (2011).

    Article  Google Scholar 

  14. F. Mehdi, T. G. Johansson, C. M. White, and J. W. Naughton, On determining wall shear stress in spatially developing two-dimensional wall-bounded flows, Exp. Fluids 55, 1656 (2014).

    Article  Google Scholar 

  15. Z. Xia, P. Zhang, and X. I. A. Yang, On skin friction in wall-bounded turbulence, Acta Mech. Sin. 37, 589 (2021), arXiv: 2007.14619.

  16. W. Li, Y. Fan, D. Modesti, and C. Cheng, Decomposition of the mean skin-friction drag in compressible turbulent channel flows, J. Fluid Mech. 875, 101 (2019).

    Article  MathSciNet  Google Scholar 

  17. S. Pirozzoli, Decomposition of the mean friction drag in zero-pressure-gradient turbulent boundary layers, Phys. Fluids 31, 086105 (2019).

    Article  Google Scholar 

  18. K. Fukagata, K. Iwamoto, and N. Kasagi, Novel turbulence control strategy for simulta-neously achieving friction drag reduction and heat transfer augmentation, in: Proceedings of 4th International Symposium Turbulence and Shear Flow Phenomena (Begell House Digital Library, Williamsburg, 2005) pp. 307–312.

    Google Scholar 

  19. J. Liu, P. Zhao, M. Lei, S. Yang, and H. Nemati, Numerical investigation of spatial-developing turbulent heat transfer in forced convections at different supercritical pressures, Int. J. Heat Mass Transfer. 159, 120128 (2020).

    Article  Google Scholar 

  20. Y. Morinishi, S. Tamano, and K. Nakabayashi, Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls, J. Fluid Mech. 502, 273 (1999).

    Article  Google Scholar 

  21. L. Duan, I. Beekman, and M. P. Martín, Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature, J. Fluid Mech. 655, 419 (2010).

    Article  Google Scholar 

  22. Y. S. Zhang, W. T. Bi, F. Hussain, X. L. Li, and Z. S. She, Mach-Number-Invariant Mean-Velocity Profile of Compressible Turbulent Boundary Layers, Phys. Rev. Lett. 109, 054502 (2012).

    Article  Google Scholar 

  23. E. J. Hopkins, and M. Inouye, An evaluation of theories for predicting turbulent skin friction andheat transfer on flat plates at supersonic and hypersonic Mach numbers, AIAA J. 9, 993 (1971).

    Article  Google Scholar 

  24. P. G. Huang, G. N. Coleman, and P. Bradshaw, Compressible turbulent channel flows: DNS results and modelling, J. Fluid Mech. 305, 185 (1995).

    Article  Google Scholar 

  25. S. Ghosh, H. Foysi, and R. Friedrich, Compressible turbulent channel and pipe flow: similarities and differences, J. Fluid Mech. 648, 155 (2010).

    Article  Google Scholar 

  26. P. Zhang, and Z. Xia, Contribution of viscous stress work to wall heat flux in compressible turbulent channel flows, Phys. Rev. E 102, 043107 (2020).

    Article  Google Scholar 

  27. D. Sun, Q. Guo, X. Yuan, H. Zhang, C. Li, and P. Liu, A decomposition formula for the wall heat flux of a compressible boundary layer, Adv. Aerodyn. 3, 33 (2021), arXiv: 2106.10968.

  28. A. Mittal, and S. S. Girimaji, Mathematical framework for analysis of internal energy dynamics and spectral distribution in compressible turbulent flows, Phys. Rev. Fluids 4, 042601(R) (2019).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Mechanics, Zhejiang University, Hangzhou, 310027, China

    Peng Zhang  (张朋), Yubin Song  (宋余滨) & Zhenhua Xia  (夏振华)

  2. State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, 310027, China

    Zhenhua Xia  (夏振华)

Authors
  1. Peng Zhang  (张朋)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yubin Song  (宋余滨)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhenhua Xia  (夏振华)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhenhua Xia  (夏振华).

Additional information

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11822208, 11772297, and 91852205) and Guangdong provincial Key Laboratory (Grant No. 2019B20203001).

Electronic Supplementary Material

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, P., Song, Y. & Xia, Z. Exact mathematical formulas for wall-heat flux in compressible turbulent channel flows. Acta Mech. Sin. 38, 321403 (2022). https://doi.org/10.1007/s10409-021-09014-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10409-021-09014-2

Keywords

Search

Navigation