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12.4.2 RNG $k$- $\epsilon$ Model



Overview


The RNG $k$- $\epsilon$ model was derived using a rigorous statistical technique (called renormalization group theory). It is similar in form to the standard $k$- $\epsilon$ model, but includes the following refinements:

These features make the RNG $k$- $\epsilon$ model more accurate and reliable for a wider class of flows than the standard $k$- $\epsilon$ model.

The RNG-based $k$- $\epsilon$ turbulence model is derived from the instantaneous Navier-Stokes equations, using a mathematical technique called "renormalization group'' (RNG) methods . The analytical derivation results in a model with constants different from those in the standard $k$- $\epsilon$ model, and additional terms and functions in the transport equations for $k$ and $\epsilon$. A more comprehensive description of RNG theory and its application to turbulence can be found in [ 59].



Transport Equations for the RNG $k$- $\epsilon$ Model


The RNG $k$- $\epsilon$ model has a similar form to the standard $k$- $\epsilon$ model:

 \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\par... ...artial x_j}\right) + G_k + G_b - \rho \epsilon - Y_M + S_k (12.4-4)

and
 \frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partia... ...on} \rho \frac{\epsilon^2}{k} - R_{\epsilon} + S_{\epsilon} (12.4-5)

In these equations, $G_k$ represents the generation of turbulence kinetic energy due to the mean velocity gradients, calculated as described in Section  12.4.4. $G_b$ is the generation of turbulence kinetic energy due to buoyancy, calculated as described in Section  12.4.5. $Y_M$ represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, calculated as described in Section  12.4.6. The quantities $\alpha_k$ and $\alpha_{\epsilon}$ are the inverse effective Prandtl numbers for $k$ and $\epsilon$, respectively. $S_k$ and $S_{\epsilon}$ are user-defined source terms.



Modeling the Effective Viscosity


The scale elimination procedure in RNG theory results in a differential equation for turbulent viscosity:

 d \Biggl(\frac{\rho^2 k}{\sqrt{\epsilon \mu}} \Biggr) = 1.72 \frac{\hat{\nu}}{\sqrt{{\hat{\nu}}^3-1+C_\nu}} d{\hat{\nu}} (12.4-6)

where
$\displaystyle \hat{\nu}$ $\textstyle =$ $\displaystyle \mu_{\rm eff}/\mu$  
$\displaystyle C_\nu$ $\textstyle \approx$ $\displaystyle 100$  

Equation  12.4-6 is integrated to obtain an accurate description of how the effective turbulent transport varies with the effective Reynolds number (or eddy scale), allowing the model to better handle low-Reynolds-number and near-wall flows .

In the high-Reynolds-number limit, Equation  12.4-6 gives
 \mu_{t} = \rho C_\mu \frac{k^2}{\epsilon} (12.4-7)

with $C_{\mu} = 0.0845$, derived using RNG theory. It is interesting to note that this value of $C_{\mu}$ is very close to the empirically-determined value of 0.09 used in the standard $k$- $\epsilon$ model.

In FLUENT, by default, the effective viscosity is computed using the high-Reynolds-number form in Equation  12.4-7. However, there is an option available that allows you to use the differential relation given in Equation  12.4-6 when you need to include low-Reynolds-number effects.



RNG Swirl Modification


Turbulence, in general, is affected by rotation or swirl in the mean flow. The RNG model in FLUENT provides an option to account for the effects of swirl or rotation by modifying the turbulent viscosity appropriately. The modification takes the following functional form:

 \mu_t = \mu_{t0} \;\; f \! \left( \alpha_s, \Omega, \frac{k}{\epsilon} \right) (12.4-8)

where $\mu_{t0}$ is the value of turbulent viscosity calculated without the swirl modification using either Equation  12.4-6 or Equation  12.4-7. $\Omega$ is a characteristic swirl number evaluated within FLUENT, and $\alpha_s$ is a swirl constant that assumes different values depending on whether the flow is swirl-dominated or only mildly swirling. This swirl modification always takes effect for axisymmetric, swirling flows and three-dimensional flows when the RNG model is selected. For mildly swirling flows (the default in FLUENT), $\alpha_s$ is set to 0.07. For strongly swirling flows, however, a higher value of $\alpha_s$ can be used.



Calculating the Inverse Effective Prandtl Numbers


The inverse effective Prandtl numbers, $\alpha_k$ and $\alpha_{\epsilon}$, are computed using the following formula derived analytically by the RNG theory:

 \left \vert \frac{\alpha - 1.3929}{\alpha_0 - 1.3929}\right... ...right\vert ^{0.3679} = \frac{\mu_{\rm mol}}{\mu_{\rm eff}} (12.4-9)

where $\alpha_0 = 1.0$. In the high-Reynolds-number limit ( $\mu_{\rm mol}/\mu_{\rm eff} \ll 1$), $\alpha_k = \alpha_{\epsilon} \approx 1.393$.



The $R_{\epsilon}$ Term in the $\epsilon$ Equation


The main difference between the RNG and standard $k$- $\epsilon$ models lies in the additional term in the $\epsilon$ equation given by

 R_{\epsilon} = \frac{C_\mu \rho \eta^3 (1-\eta/\eta_0)}{1+\beta\eta^3} \frac{\epsilon^2}{k} (12.4-10)

where $\eta \equiv Sk/\epsilon$, $\eta_0 = 4.38$, $\beta = 0.012$.

The effects of this term in the RNG $\epsilon$ equation can be seen more clearly by rearranging Equation  12.4-5. Using Equation  12.4-10, the third and fourth terms on the right-hand side of Equation  12.4-5 can be merged, and the resulting $\epsilon$ equation can be rewritten as
 \frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partia... ...n} G_b \right) - C_{2\epsilon}^* \rho \frac{\epsilon^2}{k} (12.4-11)

where $C_{2\epsilon}^*$ is given by
 C_{2\epsilon}^* \equiv C_{2\epsilon} + {C_\mu \eta^3 (1-\eta/\eta_0)\over 1+\beta\eta^3} (12.4-12)

In regions where $\eta < \eta_0$, the $R$ term makes a positive contribution, and $C_{2\epsilon}^*$ becomes larger than $C_{2\epsilon}$. In the logarithmic layer, for instance, it can be shown that $\eta \approx 3.0$, giving $C_{2\epsilon}^* \approx 2.0$, which is close in magnitude to the value of $C_{2\epsilon}$ in the standard $k$- $\epsilon$ model (1.92). As a result, for weakly to moderately strained flows, the RNG model tends to give results largely comparable to the standard $k$- $\epsilon$ model.

In regions of large strain rate ( $\eta > \eta_0$), however, the $R$ term makes a negative contribution, making the value of $C_{2\epsilon}^*$ less than $C_{2\epsilon}$. In comparison with the standard $k$- $\epsilon$ model, the smaller destruction of $\epsilon$ augments $\epsilon$, reducing $k$ and, eventually, the effective viscosity. As a result, in rapidly strained flows, the RNG model yields a lower turbulent viscosity than the standard $k$- $\epsilon$ model.

Thus, the RNG model is more responsive to the effects of rapid strain and streamline curvature than the standard $k$- $\epsilon$ model, which explains the superior performance of the RNG model for certain classes of flows.



Model Constants


The model constants $C_{1\epsilon}$ and $C_{2\epsilon}$ in Equation  12.4-5 have values derived analytically by the RNG theory. These values, used by default in FLUENT, are

C_{1\epsilon} = 1.42, \; \; C_{2\epsilon} = 1.68


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