# A Two-Time-Scale Turbulence Model and Its Application in Free Shear Flows
## 3 The \(k\)-\(\varepsilon\)-\(\tau\) Model of Turbulence
### Local Equilibrium  
In the case of local equilibrium where \(P_{k}\equiv\epsilon\), the z-transport equation becomes \(z_{1}=(\epsilon\tau/k)_{1}\):  
\[\left(\frac{dz}{dt}\right)_{1}=\frac{\tau}{k}\frac{d\epsilon}{dt}+\frac{ \epsilon}{k}\frac{d\tau}{dt}=0 \tag{19}\]  
Together with the \(\epsilon\)- and \(\tau\)-transport equations:  
\[\frac{d\epsilon}{dt}=(C_{\epsilon 1}-1)\frac{P_{k}}{\tau} \tag{20}\]  
\[\frac{d\tau}{dt}=C_{\tau 0}-C_{\tau 1}z_{1}-C_{\tau 2}z_{1} \tag{21}\]  
we obtain another relation between the coefficients. (Note that the RHS of \(\tau\)-equation is constant.)  
\[1-C_{\epsilon 1}=C_{\tau 0}-z_{1}(C_{\tau 1}+C_{\tau 2}) \tag{22}\]  
Here, one must note that we have the freedom of choosing z\({}_{0}\), provided that we initially adjust the coefficients in the \(\epsilon\)-equation accordingly. Comparison of \(\epsilon\)-equations for the \(k\)-\(\epsilon\) and \(k\)-\(\epsilon\)-\(\tau\) models yields that for \(z_{0}=6/11\), \(C_{\epsilon 1}\) becomes \(C_{\epsilon 1}\cong 0.75\), which is a good choice. This also implies that \(C_{\epsilon 2}\) can be taken as \(C_{\epsilon 2}=1.0\), and \(C_{\tau 0}=1.054\), and \(C_{\tau 1}=1.1\) are taken in parallel with the WFC model, and \(C_{\tau 2}\) is established quite well at \(C_{\tau 2}=0.59\) for the jet flow computations presented in the next section. This is also in agreement with the previous constraint (Equation (18)) on \(C_{\tau 2}\), which suggested that \(C_{\tau 2}>0.375\). A rough estimation of \(z_{1}\) from Equation (22) yields that it is in the range of \(0.47<z_{1}<0.54\) for \(P_{k}/\epsilon<1.3\). Equation (16) can be re-organized to yield:  
\[z_{\epsilon}=\frac{z_{0}+C_{z1}P_{k}/\epsilon}{1+C_{z2}P_{k}/\epsilon} \tag{23}\]  
where \(C_{z1}=C_{\epsilon 1}/(C_{\tau 1}-1)\) and \(C_{z2}=(C_{\tau 2}+1)/(C_{\tau 1}-1)\). Equation (23) has an asymptotic value of \(0.472\) for \(z_{\epsilon}\) vs. \(P_{k}/\epsilon\), with the values of coefficients given in Table 1, which indicates that the assumption of \(z_{\epsilon}\cong z_{0}=0.54\) is well within the 13% error range.  
The near-wall equilibrium layer can help in providing a relation for \(\sigma_{\epsilon}\), as in the \(k-\epsilon\) turbulence model. The \(\epsilon\)-equation can be expressed as follows by neglecting convection, diffusion of energy, and convective transport of \(\epsilon\):  
\[0=\frac{\partial}{\partial y}\bigg{(}\frac{\mu_{\tau}}{\sigma_{\epsilon}} \frac{\partial\epsilon}{\partial y}\bigg{)}+C_{\epsilon 1}\rho\frac{P_{k}}{ \tau}-C_{\epsilon 2}\rho\frac{\epsilon}{\tau} \tag{24}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \(C_{\epsilon 1}\) & \(C_{\epsilon 2}\) & \(C_{\epsilon 3}\) & \(C_{\tau 0}\) & \(C_{\tau 1}\) & \(C_{\tau 2}\) & \(C_{\tau 3}\) & \(\sigma_{k}\) & \(\sigma_{\epsilon}\) & \(\sigma_{\tau}\) \\ \hline
0.75 & 1.05 & 0.67 & 1.054 & 1.1 & 0.59 &With the assumption of constant wall shear stress and logarithmic law of the wall theory, Equation (24) can be conveyed as (here \(\tau=z_{1}\,\kappa y/C_{\mu}^{1/2}U_{\tau}\) was also employed since \(\tau=z_{1}k/\varepsilon\) is in local equilibrium):  
\[\sigma_{e}=\frac{z_{1}\kappa^{2}}{C_{\mu}^{1/2}(C_{\epsilon 2}-C_{\epsilon 1})} \tag{25}\]  
\(\sigma_{e}\) and \(\sigma_{\tau}\), together with other coefficients, are then optimized by numerical experiments to give the best results in simple shear flows.