# A Two-Time-Scale Turbulence Model and Its Application in Free Shear Flows
## 4 Model Validation  
The current section presents free shear flow implementations of the \(k\)-\(\epsilon\)-\(\tau\) model. These computations are discussed here initially to test and validate the novel model in well-documented experimental flow classes. The analysis of comparisons between estimations and measurements will provide direct evidence of flaws in turbulence models. Furthermore, free shear flows are regarded as computationally cheaper to model because of their parabolic form. As a result, numerical errors have less of an impact on predictions. Comparisons of predictions and experiments for three turbulent shear layers are performed as a validity test. These are axisymmetric and plane jets discharging into still air and plane wakes in the absence of pressure gradients. The fundamental equations that characterize the examined flows can be formulated as follows:  
* U-momentum \[\frac{\partial\rho U^{2}}{\partial x}+\frac{1}{r}\frac{\partial}{\partial y }(\rho rUV)=\frac{1}{r}\frac{\partial}{\partial y}\big{(}r\sigma_{xy}\big{)}\] (26)
* Continuity \[\frac{\partial\rho U}{\partial x}+\frac{1}{r}\frac{\partial}{\partial y}( \rho rV)=0\] (27)
* Scalar entity \[\frac{\partial\rho U\varphi}{\partial x}+\frac{1}{r}\frac{\partial}{\partial y }(\rho rV\varphi)=-\frac{1}{r}\frac{\partial}{\partial y}\big{(}r\rho_{\varphi,y}\big{)}+S_{\varphi}\] (28) where the flux is given by \(J_{\varphi,y}=-\Gamma_{\varphi}\partial\varphi/\partial y\), the secondary source term is symbolized by \(S_{\varphi}\), and \"\(\tau\)\" equals 1 for planar cases and for radial co-ordinate of axisymmetric cases, \(y\equiv r\). The stress \(\sigma_{xy}\) is given as \(\sigma_{xy}=\mu(\partial U/\partial y)-\rho\overline{uv}\) in plane geometry and \(\sigma_{xy}=\sigma_{xr}=\mu(\partial U/\partial r)-\rho\overline{uv}\) in axisymmetric geometry. Equations for turbulence models used for comparison can also be written in a similar manner and can be found in the relevant literature given at the end of the manuscript (for \(k\)-\(\epsilon\)[29], for RNG \(k\)-\(\epsilon\)[30], for SST \(k\)-\(\omega\)[31], for Realizable \(k\)-\(\epsilon\)[32]). The discretized versions of the above equations were solved by using OpenFOAM CFD software (v8) [33]. A fully conservative finite-volume method with an implicit formulation is the basis of discretization. A second order discretization scheme requiring gradient information (linearUpwind) is used for the advection of momentum and a central differencing scheme (linear) is used for the rest of the entities to approximate the cross-flow transport. The following benchmark cases have been selected in the context of this study: plane jet, round (axisymmetric) jet, and plane far wake. A plane jet involves the continuous discharge of a fluid (usually air or water) from a rectangular nozzle featuring a significant aspect ratio into a quiescent environment. The jet emerges as a continuous, planar flow with a velocity greater than the surrounding fluid. Round jets, often referred to as axisymmetric jets, represent a fundamental and extensively studied configuration in fluid mechanics. They are characterized by the flow of fluid discharged from a circular orifice into a surrounding medium, exhibiting rotational symmetry around the jet's axis. The flows are often axisymmetric and uniform in the jet core region, simplifying the flow geometry for numerical simulations [34]. Thus, only half of the shear layer was considered. The 2D wake flow is characterized by the formation of vortices or eddies trailing an object moving steadily in a fluid. The flow near the obstacle is generally complex and strongly depends on the shape of the obstacle. In the far wake region, where the flow has sufficiently developed away from the object, the wake exhibits a two-dimensional nature, allowing for simplified analysis and modeling.