# A Two-Time-Scale Turbulence Model and Its Application in Free Shear Flows
## 4 Model Validation
### Plane Jet Results  
Some of the early studies, by use of a hot wire anemometry, were performed by various researchers [35; 36; 37; 38; 39; 40]. Simulation parameters such as boundary conditions (e.g., Re = 16,500) are similar to the work of [PERSON] [41]. According to the findings, jets that are released into stagnant environments eventually form self-similar states in which their time-averaged quantities may be expressed by only one length scale and one velocity. In other words, Reynolds stress and the mean velocity distribution must not depend on the streamwise coordinate \"\(x\)\", as these scales are used to normalize them. A turbulent flow is said to be self-similar when some or all of its normalized statistical properties, such as half-width of the jet (spreading rate \(=\partial y_{h}/\partial x\)), become constant. Mean axial velocity distribution achieves self-similarity before the turbulence entities. However, there is no consensus on where this state is reached; it varies from \(x/\mathrm{d}=40\) to 100. The calculated spreading rates exhibit a high degree of concordance with a value of \(\partial y_{h}/\partial x\) of 0.109. TKE in the vicinity of the nozzle tip increases depending on its high production rate in regions characterized by a large velocity gradient. This high TKE produced in the high shear area diffuses towards the jet's center and edge as it expands. The turbulence quantity measurements are widely dispersed. The LDA measurement of [PERSON] and [PERSON] [42] gave the second lowest maximum shear stress level of \(\overline{w}_{max}/U_{cl}^{2}=\)0.02, with the highest spreading rate of 0.112 (Table 2). For jets in stagnant environments, the self-similar form is regarded as universal and independent of the jet's initial conditions. Consequently, a uniform TKE level (\(k_{j}=1/2(U_{0}I)^{2}\)) and a flat velocity profile at the jet nozzle are selected as initial conditions, where \(I\) denotes turbulence intensity. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Investigator** & **Spreading Rate** & \(\overline{w}_{max}/U_{cl}^{2}\) & **Remarks** \\ \hline [PERSON] [35] & 0.109 & 0.024 & HWA \\ [PERSON] and [PERSON] [36] & 0.11 & 0.024 & HWA \\ [PERSON] and [PERSON] [37] & 0.097 & 0.025 & CTA \\ [PERSON] [39] & 0.095 & - & HWA \\ [PERSON] [40] & 0.11 & 0.020 & HWA \\ [PERSON] and [PERSON] [43] & 0.09–0.11 & 0.019 & CTA \\ [PERSON] and [PERSON] [42] & 0.112 & 0.02 & LDA \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of experimental and model results for incompressible plane jet. The computed mean velocity distribution for plane jets in the area of self-similarity is displayed in Figure 2 in the axial direction. It is assessed in comparison with the experimental results of [PERSON] [38], and predictions of the other two-equation models considered in the current manuscript. Figure 2 presents a comparison of the shear stress \"\(\overline{uv}\)\" and TKE data from [PERSON] and [PERSON] [36] and [PERSON] [35], respectively. The modified \(k\)-\(\epsilon\)-\(\tau\) model provides better agreement for shear stress and complies with TKE patterns observed in both experimental findings documented in the scientific literature closer to the centre of the jet, and slightly underestimates shear stress towards the distance of \(2y_{h}\). Table 2 displays the spreading-rates as well as the maximum shear stress levels achieved for the plane jet in comparison to different experimental results and values determined using various two-equation models. It is evident that the \(k\)-\(\epsilon\)-\(\tau\) model predicts \begin{table}
\begin{tabular}{c c c} \hline \hline
**Investigator** & **Spreading Rate** & \(\overline{uv}_{max}/\mathbf{U}_{el}^{2}\) & **Remarks** \\ \hline \(k\)–\(\epsilon\) model & 0.108 & 0.022 \\ SST \(k\)–\(\omega\) model & 0.113 & 0.0215 \\ RNG \(k\)–\(\epsilon\) model & 0.117 & 0.029 \\ \(k\)–\(\epsilon\)–\(\tau\) model & 0.109 & 0.0216 \\ \hline \hline \end{tabular}
\end{table}