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12.4.1 Standard $k$- $\epsilon$ Model



Overview


The simplest "complete models'' of turbulence are two-equation models in which the solution of two separate transport equations allows the turbulent velocity and length scales to be independently determined. The standard $k$- $\epsilon$ model in FLUENT falls within this class of turbulence model and has become the workhorse of practical engineering flow calculations in the time since it was proposed by Launder and Spalding [ 196]. Robustness, economy, and reasonable accuracy for a wide range of turbulent flows explain its popularity in industrial flow and heat transfer simulations. It is a semi-empirical model, and the derivation of the model equations relies on phenomenological considerations and empiricism.

As the strengths and weaknesses of the standard $k$- $\epsilon$ model have become known, improvements have been made to the model to improve its performance. Two of these variants are available in FLUENT: the RNG $k$- $\epsilon$ model [ 408] and the realizable $k$- $\epsilon$ model [ 330].

The standard $k$- $\epsilon$ model [ 196] is a semi-empirical model based on model transport equations for the turbulence kinetic energy ( $k$) and its dissipation rate ( $\epsilon$). The model transport equation for $k$ is derived from the exact equation, while the model transport equation for $\epsilon$ was obtained using physical reasoning and bears little resemblance to its mathematically exact counterpart.

In the derivation of the $k$- $\epsilon$ model, the assumption is that the flow is fully turbulent, and the effects of molecular viscosity are negligible. The standard $k$- $\epsilon$ model is therefore valid only for fully turbulent flows.



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


The turbulence kinetic energy, $k$, and its rate of dissipation, $\epsilon$, are obtained from the following transport equations:

 \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\par... ...artial x_j}\right] + G_k + G_b - \rho \epsilon - Y_M + S_k (12.4-1)

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

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. $C_{1 \epsilon}$, $C_{2 \epsilon}$, and $C_{3 \epsilon}$ are constants. $\sigma_k$ and $\sigma_{\epsilon}$ are the turbulent Prandtl numbers for $k$ and $\epsilon$, respectively. $S_k$ and $S_{\epsilon}$ are user-defined source terms.



Modeling the Turbulent Viscosity


The turbulent (or eddy) viscosity , $\mu_t$, is computed by combining $k$ and $\epsilon$ as follows:

 \mu_t = \rho C_{\mu} \frac{k^2}{\epsilon} (12.4-3)

where $C_{\mu}$ is a constant.



Model Constants


The model constants $C_{1 \epsilon}, C_{2 \epsilon}, C_{\mu}, \sigma_k,$ and $\sigma_{\epsilon}$ have the following default values [ 196]:

C_{1 \epsilon} = 1.44, \;\; C_{2 \epsilon} = 1.92, \;\; C_{\mu} = 0.09, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.3

These default values have been determined from experiments with air and water for fundamental turbulent shear flows including homogeneous shear flows and decaying isotropic grid turbulence. They have been found to work fairly well for a wide range of wall-bounded and free shear flows.

Although the default values of the model constants are the standard ones most widely accepted, you can change them (if needed) in the Viscous Model panel.


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