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


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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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 :


q :

heat source

R i :

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

Sc t :

turbulent Schimdt

s ij :

stress rate tensor

T :


t :


U :

internal energy

u i :

Eulerian velocity

x i :


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 :


υ t :

turbulent viscosity


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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).

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