# A Two-Time-Scale Turbulence Model and Its Application in Free Shear Flows
## 5 Conclusions  
In this study, a novel three-equation turbulence model was developed based on the limitations of the WFC model for predicting the behavior of shear flows, and its performance was then presented for plane and axisymmetric jets, and plane asymptotic wake.  
The findings indicate that the \(k\)-\(\varepsilon\)-\(\tau\) model outperforms the \(k\)-\(\varepsilon\) model by a significant margin. The first benefit is that the \"plane jet/round jet\" anomaly is better resolved by avoiding additional difficulties of second moment closures. The second benefit is that the flow behavior is better predicted than with the two-equation models in the wake, which is characterized by its weak shear form. Furthermore, it was discovered that:  
* Consistent with the measurements, the three-equation model (utilizing identical parameters across all three cases) estimates a spread rate of 0.109 for the plane jet;  
Figure 9: (**a**) Normalized turbulence viscosities across the plane jet for a range of “f”. (**b**) Turbulence viscosity components for a range of “f” that originates from two distinct length scales.  
* The model estimates the round jet spreading rate of 0.089, which is over 23% better than the \(k\)-\(\varepsilon\) and SST \(k\)-\(\omega\) models and consistent with the experimental data;
* The parameter for the spreading of the plane wake is estimated to be 0.081, which is approximately 6% more accurate than the \(k\)-\(\varepsilon\) model and 4% more accurate than the SST \(k\)-\(\omega\) model;
* Time scales \(\tau\) and \(k/\varepsilon\) behave quite similar to each other in most parts of the jet, as expected (in non-equilibrium situations, as in the compression stroke of an IC engine, these two will differ considerably);
* For the turbulence viscosity, several options, such as the geometric and arithmetic averages with a weighting factor, were investigated and shown to have no significant advantage over the traditional one for the types of flows tested.  
Conceptualization, M.Z.G. and M.U.Y.; methodology, M.Z.G. and M.U.Y.; software, M.U.Y.; validation, M.Z.G. and M.U.Y.; formal analysis, M.Z.G. and M.U.Y.; data curation, M.Z.G., M.U.Y. and S.S.; writing--original draft preparation, M.Z.G., M.U.Y. and S.S.; writing--review and editing, M.Z.G. and M.U.Y.; visualization, M.U.Y. and S.S.; supervision, M.Z.G.; project administration, M.Z.G.; funding acquisition, M.Z.G. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Scientific Research Projects Commission (BAPKO) of Marmara University under grant number FEN-C-DRP 131217-0677. This support is gratefully acknowledged.  
Not applicable.  
All benchmark data utilized in the study have been meticulously detailed and tabulated across various sections. These tables comprehensively present the benchmark data used for comparison, analysis, and validation throughout our research. Furthermore, to ensure transparency and facilitate further inquiry, detailed references citing the sources of these benchmark datasets have been diligently included in the References section (References [35; 36; 37; 38; 39; 40; 41; 42; 45; 46; 48; 49; 50; 51; 52; 54; 55; 56; 57; 58; 59; 60]).  
The authors declare no conflicts of interest.