RANS Turbulent Heat Transfer

For RANS turbulence models, the definition of the mean heat flux in the energy equation is based on a Boussinesq approximation. For the SKE LRe model, Simcenter STAR-CCM+ provides the Temperature Flux model, which replaces the Boussinesq approximation by an algebraic formulation for the turbulent heat flux.

By default, the mean heat flux $\bar{q}$ in the energy equation Eqn. (1145) is assumed to be proportional to the turbulent eddy viscosity as:

Figure 1. EQUATION_DISPLAY

\bar{q} = - (κ + \frac{μ_{t} C_{p}}{\Pr_{t}}) \nabla \bar{T}

(1343)

where:

$κ$ is the thermal conductivity of the fluid.
$μ_{t}$ is the turbulent eddy viscosity as given by the respective turbulence model. For RST models, $μ_{t}$ is redefined as given by Eqn. (1340).
$C_{p}$ is the specific heat.
$\Pr_{t}$ is the turbulent Prandtl number.
$\bar{T}$ is the mean temperature.

However, this assumption fails when buoyancy forces are dominant, or at locations very near the wall. A remedy proposed by Kenjeres and others [346] is to replace the Boussinesq approximation by an algebraic formulation for the turbulent heat flux itself. This formulation is a function of the Reynolds-stress anisotropy and the temperature variance, for which an additional transport equation is solved. The performance of the algebraic Temperature Flux model is strongly linked to the correct approximation of the near-wall turbulent behavior, thus requiring a low-Reynolds number turbulence model.

For the Temperature Flux model, the heat flux is defined as:

Figure 2. EQUATION_DISPLAY

\bar{q} = - κ \nabla \bar{T} - {ρ C}_{p} \bar{v θ}

(1344)

where $ρ$ is the density.

The algebraic formulation for the turbulent heat flux $\overline{v θ}$ is given as:

Figure 3. EQUATION_DISPLAY

\overline{v θ} = {- C}_{{t u}_{0}} \frac{k}{ε} (C_{{t u}_{1}} R \cdot \nabla \bar{T} + C_{{t u}_{2}} \nabla \bar{v} \cdot \bar{v θ} + C_{{t u}_{3}} β \overline{θ^{2}} g) + C_{{t u}_{4}} A \cdot \bar{v θ}

(1345)

where:

$C_{{t u}_{0}}$ , $C_{{t u}_{1}}$ , $C_{{t u}_{2}}$ , $C_{{t u}_{3}}$ , and $C_{{t u}_{4}}$ are Model Coefficients.
$k$ is the turbulent kinetic energy.
$ε$ is the turbulent dissipation rate.
$R$ is the Reynolds stress tensor given by Eqn. (1308).
$β$ is the thermal expansion coefficient.
$g$ is the gravity vector.

$A$ is the Reynolds stress anisotropy tensor given as:

Figure 4. EQUATION_DISPLAY

A = \frac{1}{k} R - \frac{2}{3} I

(1346)

where $I$ is the identity tensor.

The temperature variance $\bar{θ^{2}}$ is computed by solving an additional transport equation:

Figure 5. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ \bar{θ^{2}}) + \nabla\cdot (ρ \bar{θ^{2}} \bar{v}) = \nabla\cdot [(μ + \frac{μ_{t}}{σ_{θ^{2}}}) \nabla \bar{θ^{2}}] + G_{θ} - ρ ε_{θ}

(1347)

where:

$μ$ is the dynamic viscosity of the fluid.
$σ_{θ^{2}}$ is a Model Coefficient.

The production of the temperature variance $G_{θ}$ is defined as:

Figure 6. EQUATION_DISPLAY

G_{θ} = - ρ \bar{v θ} \cdot \nabla \bar{T}

(1348)

The temperature variance dissipation rate $ε_{θ}$ is obtained from the definition of the thermal time scale $T_{θ}$ as:

Figure 7. EQUATION_DISPLAY

T_{θ} = \frac{\overline{θ^{2}}}{2 ε_{θ}}

(1349)

and the assumption of a constant turbulent-to-thermal time-scale ratio:

Figure 8. EQUATION_DISPLAY

{R = T}_{θ} / T

(1350)

where:

$R$ is a Model Coefficient.
$T$ is the turbulent time scale given by Eqn. (1164) or Eqn. (1165), respectively.

The turbulent heat flux $\overline{v θ}$ is also used to set the buoyancy production term $G_{b}$ in the transport equations for $k$ and $ε$ as:

Figure 9. EQUATION_DISPLAY

G_{b} = - ρ β g \cdot \bar{v θ}

(1351)

Model Coefficients


$R$	$σ_{θ^{2}}$
0.5	1.0


$C_{{t u}_{0}}$	$C_{{t u}_{1}}$	$C_{{t u}_{2}}$	$C_{{t u}_{3}}$	$C_{{t u}_{4}}$
0.12	1.0	0.6	0.6	1.5