Reynolds-Averaged Navier-Stokes Equations Describing Turbulent Flow and Heat Transfer Behavior for Supercritical Fluid

Abstract

Supercritical fluid has been widely applied in many industrial applications. The traditional Reynolds-averaged Navier-Stokes (RANS) equations are directly applied for turbulent flow and heat transfer of the supercritical fluid, ignoring turbulent effect of the thermal physical properties due to the intense nonlinearity. This paper deduces a set of Reynolds-averaged Navier-Stokes equations for supercritical fluid (SCF-RANS equations) to depict turbulent flow and heat transfer of the supercritical fluid taking all the physical parameters as variables. The SCF-RANS equations include many new correlation terms due to fluctuation of the thermal physical properties. Model methods for the new correlation term have been discussed for closing the SCF-RANS equations. Some of them have relatively mature models, while others are completely new and need profound physical theoretical analysis for proposing reasonable models. This paper provides referable information for these new correlations as far as authors know. The SCF-RANS equations not only provide the formulation special for flow and heat transfer of the supercritical fluid, but also represent the most sophisticate form of the RANS equations, for every involved physical property has been considered as variable without any simplification.

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Abbreviations

C s :

model constant in Eqs. (30) and (31)

\(C_s^\rho \) :

model constant in Eqs. (32) and (33)

c v :

constant-volume specific heat

c 0 :

positive constant

D t :

isotropic turbulent diffusivity tensor

\(D_{ij}^t\) :

anisotropic turbulent diffusivity tensor

g j :

gravity acceleration

J ij :

molecular diffusion component in Eq. (27)

k :

thermal conductivity; turbulent kinetic energy

P i :

production term in Eq. (27)

P ij :

production term in Reynolds stress equation

p :

pressure

q :

heat source

R i :

pressure-scalar-gradient term in Eq. (27)

Sc t :

turbulent Schimdt

s ij :

stress rate tensor

T :

temperature

t :

time

U :

internal energy

u i :

Eulerian velocity

x i :

direction

Γ :

model constant in Eq. (36)

Γ t :

model constant in Eq. (37)

δ ij :

Kronecker delta

ε :

turbulent kinetic dissipation rate

ε i :

scalar-flux dissipation term in Eq. (27)

ε ij :

turbulent dissipation in Reynolds stress equation

μ :

kinetic viscosity

ε t :

turbulent viscosity

Π ij :

velocity-pressure gradient in Reynolds stress equation

ρ :

density

υ t :

turbulent viscosity

References

  1. [1]

    Lemmon E.W., Huber M.L., McLinden M.O., IST standard reference database 23: reference fluid thermodynamic and transport properties — REFPROP, Version 9.1, National Institute of Standards and Technology, Gaithersburg, 2013.

    Google Scholar 

  2. [2]

    Cheng X., Liu X. J., Research challenges of heat transfer to supercritical fluids, ASME Journal of Nuclear Engineering and Radiation Science, 2017, 4(1): 011003.

    Article  Google Scholar 

  3. [3]

    Shitsman M., Impairment of the heat transmission at supercritical pressures (Heat transfer process examined during forced motion of water at supercritical pressures). High Temperature, 1963, 1: 237–244.

    Google Scholar 

  4. [4]

    Yamagata K., Nishikawa K., Hasegawa S., Fujii T., Yoshida S., Forced convective heat transfer to supercritical water flowing in tubes. International Journal of Heat and Mass Transfer, 1972, 15(12): 2575–2593.

    Article  Google Scholar 

  5. [5]

    Ackerman J.W., Pseudoboiling heat transfer to supercritical pressure water in smooth and ribbed tubes. ASME Journal of Heat Transfer, 1970, 92: 490–497.

    Article  Google Scholar 

  6. [6]

    Li Y., Chen Y.W., Zhang Y.C., Sun F., Xie G. N., An improved heat transfer correlation for supercritical aviation kerosene flowing upward and downward in vertical tubes. Journal of Thermal Science, 2020, 29(4): 131–143.

    ADS  Article  Google Scholar 

  7. [7]

    Hiroaki T., Ayao T., Masaru H., Effects of buoyancy and of acceleration owing to thermal expansion on forced turbulent convection in vertical circular tubes—Criteria of the effects, velocity and temperature profiles, and reverse transition from turbulent to laminar flow. International Journal of Heat and Mass Transfer, 1973, 16(6): 1267–1288.

    Article  Google Scholar 

  8. [8]

    Jackson J.D., Hall W.B., Turbulent Forced Convection in Channels and Bundles, 2nd ed., Hemisphere, Washington, D.C., 1979, pp.: 613–640.

    Google Scholar 

  9. [9]

    McEligot D.M., Coon C.W., Perkins H.C., Relaminarization in tubes. Journal of Heat and Mass Transfer, 1970, 13(2): 431–433.

    Article  Google Scholar 

  10. [10]

    Tian R., Dai X.Y., Wang D.B., Shi L., Study of variable turbulent Prandtl number model for heat transfer to supercritical fluids in vertical tubes. Journal of Thermal Science, 2018, 27(3): 213–222.

    ADS  Article  Google Scholar 

  11. [11]

    Favre A., The mechanics of turbulence, first ed., Gordon and Breach, New York, 1964.

    Google Scholar 

  12. [12]

    Combest D.P., Ramachandran P.A., Dudukovic M.P., On the gradient diffusion hypothesis and passive scalar transport in turbulent flows. Industrial & Engineering Chemistry Research, 2011, 50(15): 8817–8823.

    Article  Google Scholar 

  13. [13]

    Batchelor G.K., Diffusion in a field of homogeneous turbulence. Mathematical Proceedings of the Cambridge Philosophical Society, 1952, 48(2): 345–362.

    Article  MATH  Google Scholar 

  14. [14]

    Daly B.J., Harlow F.H., Transport equations in turbulence. Physics of Fluids, 1970, 13(11): 2634–2649.

    ADS  Article  Google Scholar 

  15. [15]

    Fox R.O., Computational models for turbulent reacting flows, Cambridge University Press, Oxford, 2003.

    Google Scholar 

  16. [16]

    Younis B.A., Speziale C.G., Clark T.T., A rational model for the turbulent scalar fluxes. Proceedings of the Royal Society A, 2005, 461(2054): 575–594.

    Google Scholar 

  17. [17]

    Boussinesq J., Théorie de l’écoulement tourbillant. Mémoires présentés par Divers Savants à l’Acad. des Sci. Inst. Nat. France, 1877, 23: 46–50.

    Google Scholar 

  18. [18]

    Pletcher R.H., Tannehill J.C., Anderson D.A., Computational fluid mechanics and heat transfer, second ed., Taylor & Francis, Washington, D.C., 1997.

    Google Scholar 

  19. [19]

    Wilcox D.C., Turbulence modeling for CFD, third ed., DCW Industries, Canada, 2006.

    Google Scholar 

  20. [20]

    Mellor G.L., Herring H.J., A survey of the mean turbulent field closure models. AIAA Journal, 1973, 11(5): 590–599.

    ADS  Article  MATH  Google Scholar 

  21. [21]

    Hanjalic K., Launder B.E., A Reynolds stress model of trubulence and its application to thin shear flows. Journal of Fluid Mechanics, 1972, 52(4): 609–638.

    ADS  Article  MATH  Google Scholar 

  22. [22]

    Launder B.E., Simulation and modeling of turbulent flows, first ed., Oxford University Press, New York, 1996 pp. 243–310.

    Google Scholar 

  23. [23]

    Dutta A., Tarbell J.M., Closure models for turbulent reacting flows. AIChE Journal, 1989, 35(12): 2013–2027.

    Article  Google Scholar 

  24. [24]

    Heeb T.G., Brodkey R.S., Turbulent mixing with multiple second-order chemical reactions. AIChE Journal, 1990, 36(10): 1457–1470.

    Article  Google Scholar 

  25. [25]

    Shenoy U.V., Toor H.L., Unifying indicator and instantaneous reaction methods of measuring micromixing. AIChE Journal, 1990, 36(2): 227–232.

    Article  Google Scholar 

  26. [25a]

    Launder B.E., Reece G.J., Rodi W., Progress in the development of a Reynolds-stress turbulence closure. Journal of Fluid Mechanics, 1975, 68(3): 537–566.

    ADS  Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the support of National Key R&D Plan of China (2017YFB0903601), National Natural Science Foundation of China (51606186), Newton Advanced Fellowship of the Royal Society (NA170093) and Strategic Priority Research Program of the Chinese Academy of Sciences (XDA21070200).

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Correspondence to Haisheng Chen.

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Yang, Z., Cheng, X., Zheng, X. et al. Reynolds-Averaged Navier-Stokes Equations Describing Turbulent Flow and Heat Transfer Behavior for Supercritical Fluid. J. Therm. Sci. 30, 191–200 (2021). https://doi.org/10.1007/s11630-020-1339-6

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Keywords

  • SCF-RANS equations
  • supercritical fluid
  • turbulence
  • Reynolds-averaged
  • Navier-Stokes equations