# A Two-Time-Scale Turbulence Model and Its Application in Free Shear Flows
## 4 Model Validation
### Round Jet Results  
Table 3 gives a summary of experiments and simulation outcomes of the turbulence properties in the self-similar area. Compared to a plane jet, a round jet spreads slower, according to these observations, and takes between 50 and 70 diameters to achieve its self-similar form. [PERSON] and [PERSON] (1999) reported a peak value of about \(\overline{uv}/U_{cl}^{2}=0.021\) at 70 \(x\)/d and gave a value of 0.094 for the spreading rate, although early research indicates a spreading rate of \(dy_{h}/dx=0.086\). The work by [PERSON] et al. (1999) contains the simulation parameters for round jet simulations, specifically Re = 95,500 and \(U_{i}=56.2\) m/s. The traditional models consistently anticipate higher spreading rates for the axisymmetric case, also known as the \"plane jet/round jet\" anomaly, which was initially documented by [PERSON] and [PERSON] (1999) in their analysis using the \(k\)-\(k_{l}\) model. However, the round jet spreading rates predicted by the SST \(k\)-\(\omega\) and \(k\)-\(\epsilon\) models are around 28% too high, while the \(k\)-\(\epsilon\)-\(\tau\) model (applying the identical model parameters to the plane jet) provides a considerably closer value of 0.089. A realizable \(k\)-\(\epsilon\) model is close behind the \(k\)-\(\epsilon\)-\(\tau\) model in terms of spreading rates, with a value of 0.088. The predicted mean axial velocity, shear stress, and TKE profiles for axisymmetric jets are compared to the work of [PERSON] (1999) at the self-similar region in Figure 5. It is evident that consistency of the \(k\)-\(\epsilon\)-\(\tau\) model outputs are quite satisfactory. The maximum shear stress profiles are projected to be 22% higher by the \(k\)-\(\epsilon\) model; that is compatible with the prediction of its spreading rate. The \(k\)-\(\epsilon\)-\(\tau\) model, after making a slight hump which is not consistent with experiments at about 0.5 \(y_{h}\), levels with the data from the measurements of [PERSON] (1999), and [PERSON] et al. (1999); thereafter, the agreement with it is quite good. Interestingly, the aforementioned \"plane jet/round jet\" anomaly does not appear in the \(k\)-\(\epsilon\)-\(\tau\) model. Unlike the close error margin considering the plane jet, TKE profiles are overestimated by about 9% from the jet axis in the first quarter. On the other hand, TKE predictions of the SST \(k\)-\(\omega\) model are much higher than the rest of the models considered in the manuscript, in terms of both shear stress and TKE distributions. Turbulence time scale \(\tau^{*}\) level is consistently higher than the \(k\)-\(\epsilon\) level in the \(k\)-\(\epsilon\)-\(\tau\) model, up to the outer edge of the jet, by about 5% at the jet centre and 12% at 1.4 \(y_{h}\) (Figure 6), but agrees with \(k/\epsilon\) from the \(k\)-\(\epsilon\) model. It can also be observed in the same figure that the \(z\)-level is increased slightly (7-8%) compared to plane jet (Figure 3). This could be due to an increased energy dissipation rate predicted in the round jet as can be seen in the energy budget in Figure 7. It can also be seen that the convection and production terms in the energy budget are predicted reasonably well compared with the measurements. The change in sign in the diffusion term, close to the centre of the jet, is consistent with the predictions of TKE. However, this is not supported by the experiments as seen in Figures 5 and 7.