# A Two-Time-Scale Turbulence Model and Its Application in Free Shear Flows
## 4 Model Validation
### Plane Far Wake Results  
It is possible to define a normalized spreading parameter for the wake as \(S_{w}=(U_{E}/U_{o})dy_{h}/dx\), where \(U_{E}\) is the outer stream velocity and \(U_{o}\) is the velocity deficit at the centerline of the wake [53; 54]. The simulation parameters for plane far wake simulations are specified in the paper by [PERSON] et al. [54] as Re = 2800 and \(U_{i}\) = 6.7 m/s. Simulation results proved that the measurements of [PERSON] et al. [55] seemed to be the most reliable. They state that the Reynolds number calculated by using the cylinder diameter is equal to 1360 and that the similarity zone begins at \(x/D>400\) in the wake of the cylinder that has a certain diameter (D = 6.35 mm) [54]. Their similarity research suggests that the parameter for spreading must have a specific value, irrespective of the wake generator type. Furthermore, [PERSON] et al. [56] asserted that the plane wake reaches a universal, self-sustaining form that is independent of the initial body when it is sufficiently far downstream, depending on the self-preserving properties demonstrated by second order moments. However, measurements of [PERSON] et al. [55] suggested that the developing behavior and asymptotic structure of plane wake can be impacted by the form of the wake generator, and this was one of the causes of the disagreement observed by different workers.  
It is particularly challenging to estimate both weak and strong shear flows using a pair of coefficients, particularly when employing eddy-viscosity models. The phenomenon known as \"weak-strong shear flow\" is thought to be induced by the dissipation rate equation in its simulated form. Table 4 presents a compilation of experimental and projected values for the wake spreading parameters in the aftermath of the wake generator, as determined by the typical two-equation models and \(k\)-\(\epsilon\)-\(\tau\) model. Regarding the experiments of [PERSON] et al. [55], the modified \(k\)-\(\epsilon\)-\(\tau\) model underestimates this spread parameter by less than 2%, suggesting a considerably better result compared to the significant underestimation by the \(k\)-\(\epsilon\) and SST \(k\)-\(\omega\) models.  
Figure 7: Round jet energy budget [49].  
As can be seen in Figure 8, the mean axial velocity profile for the far wake is well predicted, and the result obtained is very similar to that of other models. Despite a 23% lower estimation of maximum shear stress level compared to experimental values [55], the modified \(k\)-\(\epsilon\)-\(\tau\) model yielded a better value than the k-\(\epsilon\) model, which underestimated it by 33%. This finding aligns with the comparative evaluation of the models' spreading parameter performance. Moreover, TKE profiles are predicted to be within 3% deviation close to the centre of the wake, up to the point of 1.2 \(y_{h}\). Conversely, throughout the interval of 0-1.2 \(y_{h}\), shear stress profiles are underestimated by approximately 15-30 percent, yielding a reduced level of concurrence in comparison to the other models considered in this research. When compared to experimental behavior, both turbulence energy profiles and shear stress appear to cease instantly near the outer edge.  
\begin{table}
\begin{tabular}{c c c} \hline
**Investigator** & **Spreading Parameter** & \(-\overline{uv}_{max}/\textbf{U}_{cl}^{2}\) \\ \hline [PERSON] and [PERSON] [43] & 0.096 & 0.037 \\ [PERSON] (from [PERSON] and & \\ [PERSON] [42]) & 0.089 & - \\ [PERSON] et al. [55] & 0.082 & 0.048 \\ [PERSON] [57] & 0.098 & 0.051 \\ [PERSON] and [PERSON] [58] & 0.092 & - \\ \(k\)-\(\epsilon\) model & 0.077 & 0.036 \\ SST \(k\)-\(\omega\) model & 0.079 & 0.042 \\ RNG \(k\)-\(\epsilon\) model & 0.099 & 0.046 \\ \(k\)-\(\epsilon\)-\(\tau\) model & 0.081 & 0.039 \\ \hline \end{tabular}
\end{table}
Table 4: Comparison of experimental and model results for plane far wake.