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12.4.7 Convective Heat and Mass Transfer Modeling in the $k$- $\epsilon$ Models

In FLUENT, turbulent heat transport is modeled using the concept of Reynolds' analogy to turbulent momentum transfer. The "modeled'' energy equation is thus given by the following:

 \frac{\partial}{\partial t} (\rho E) + \frac{\partial}{\par... ...al T}{\partial x_j} + u_i (\tau_{ij})_{\rm eff} \right) + S_h (12.4-29)

where $E$ is the total energy, $k_{\rm eff}$ is the effective thermal conductivity, and $(\tau_{ij})_{\rm eff}$ is the deviatoric stress tensor, defined as

(\tau_{ij})_{\rm eff} = \mu_{\rm eff} \left(\frac{\partial u_... ...} \mu_{\rm eff} \frac{\partial u_k}{\partial x_k} \delta_{ij}

The term involving $(\tau_{ij})_{\rm eff}$ represents the viscous heating, and is always computed in the density-based solvers. It is not computed by default in the pressure-based solver, but it can be enabled in the Viscous Model panel.

Additional terms may appear in the energy equation, depending on the physical models you are using. See Section  13.2.1 for more details.

For the standard and realizable $k$- $\epsilon$ models, the effective thermal conductivity is given by

k_{\rm eff} = k + \frac{c_p \mu_t}{{\rm Pr}_t}

where $k$, in this case, is the thermal conductivity. The default value of the turbulent Prandtl number is 0.85. You can change the value of the turbulent Prandtl number in the Viscous Model panel.

For the RNG $k$- $\epsilon$ model, the effective thermal conductivity is

k_{\rm eff} = \alpha c_p \mu_{\rm eff}

where $\alpha$ is calculated from Equation  12.4-9, but with $\alpha_0 = 1/{\rm Pr} = k/\mu c_p$. The fact that $\alpha$ varies with $\mu_{\rm mol}/\mu_{\rm eff}$, as in Equation  12.4-9, is an advantage of the RNG $k$- $\epsilon$ model. It is consistent with experimental evidence indicating that the turbulent Prandtl number varies with the molecular Prandtl number and turbulence [ 175]. Equation  12.4-9 works well across a very broad range of molecular Prandtl numbers, from liquid metals ( ${\rm Pr} \approx 10^{-2}$) to paraffin oils ( ${\rm Pr} \approx 10^{3}$), which allows heat transfer to be calculated in low-Reynolds-number regions. Equation  12.4-9 smoothly predicts the variation of effective Prandtl number from the molecular value ( $\alpha = 1/{\rm Pr}$) in the viscosity-dominated region to the fully turbulent value ( $\alpha = 1.393$) in the fully turbulent regions of the flow.

Turbulent mass transfer is treated similarly. For the standard and realizable $k$- $\epsilon$ models, the default turbulent Schmidt number is 0.7. This default value can be changed in the Viscous Model panel. For the RNG model, the effective turbulent diffusivity for mass transfer is calculated in a manner that is analogous to the method used for the heat transport. The value of $\alpha_0$ in Equation  12.4-9 is $\alpha_0 = 1/{\rm Sc}$, where Sc is the molecular Schmidt number.


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