Production-Term Modification

Amongst the various recent publications on one-equation models, Menters [2] rigorous derivation of a generic one-equation model casted in terms of a transport equation for the eddy-viscosity $\nu_t$ is, perhaps, the most instructive. Based on the application of local-equilibrium assumptions to a two-equation - e.g. $k-\varepsilon$ - model, Menters approach provides a keen insight into the one-equation modeling framework, in particular the related coefficients. The procedure reveals that the four most influential production and destruction terms of the two-equation approach collapse into a single production-type term in the one-equation framework, viz.

$$P_{\tilde \nu_t} = \tilde \nu_t S^* \quad \left( C_{\varepsilon 2} - C_{\varepsilon ... ...mu \, S^*k/\varepsilon ) \quad \approx \quad \tilde \nu_t \tilde S \quad C_{b1}$$ (5)

The coefficient $C_{b1}$ is thus crucial to the model's predictive performance. As indicated by eq. (5), $C_{b1}$ is a function of the strain rate and model coefficients. Substituting the employed coefficients of the background $k-\varepsilon$ turbulence model by $C_{\varepsilon1}=1.45$ and $C_{\varepsilon2}=1.9$, additionally employing $c_\mu S^*k/\varepsilon = \sqrt{c_\mu}$ one obtains $C_{b1}=\sqrt{c_\mu} (C_{\varepsilon2}-C_{\varepsilon1}) = 0.135$, which is close to the original SA model ( $C_{b1}=0.1355$). Both expressions, $\left( C_{\varepsilon 2} - C_{\varepsilon 1} \right)$ and the anisotropy parameter $c_\mu$, are itself a function of strain and rotation rate invariants. In general, they both tend to decrease with an increase of strain, which motivates the following modification of the standard coefficient $C_{b1}$:

$$C_{b1} = 0.1355 \sqrt{\Gamma} \qquad \Gamma=\min \left(1.25, \max(\gamma,0.75) \right) \; , \qquad \gamma = \max\left(\alpha_1,\alpha_2\right)$$

$$\alpha_1= \left( 1.01 \frac{ \tilde \nu_t}{ \kappa^2 \: l_n ^2 S^... ..._2= \max \left( 0,1-\tanh \left( \frac{\nu_t^+}{68} \right) \right)^{0.65} \; .$$ (6)

Due to the lack of an individual length-scale the approach relies on some heuristics based on the comparison of a mixing-length related eddy-viscosity with the value obtained from the solution of the transport equation. The modification $\sqrt{\Gamma}$ primarily causes a reduction of production for excessive strains via $\alpha_1$. Additionally, undesirable wall-damping is suppressed by the inclusion of $\alpha_2$. The limitation of $\Gamma$ are necessary due to the proportionality of $\tilde \nu_t$ and $\alpha_1$, i.e. a decreasing $\tilde \nu_t$ causes a decreasing $\alpha_1$. Since it is closely related to the destruction parameter $\Psi$, the modification of $\tilde C_{b1}$ represents a cross-term between production and destruction.