# A Two-Time-Scale Turbulence Model and Its Application in Free Shear Flows
## 2 The WFC Model of Turbulence  
The motivation behind the WFC three-equation turbulence model is to decouple the dissipation and the time and length scales and introduce a minimum of additional complexity. To this end, [PERSON], [PERSON], and [PERSON] [5] introduced a model equation for a turbulence time scale \(\tau\) to be solved together with the \(k\)- and \(\varepsilon\)-equations.  
For homogeneous isotropic decay flow, the exact equation of the TKE reduces to:  
\[\frac{dk}{dt}=-\varepsilon \tag{1}\]  
The model equation for dissipation rate is modified to include the new turbulence time scale:  
\[\frac{d\varepsilon}{dt}=-\frac{\varepsilon}{\tau} \tag{2}\]  
This equation can also be seen as the definition of the new time scale \(\tau\). In isotropic turbulence decay behind a grid, \(k\) decays according to power law \((k=t^{-n})\). The turbulence-kinetic-energy Equation (1) then yields \(\varepsilon=nt^{-n-1}\). These can be used together with Equation (2) in order to close the set of model equations to yield \(d\tau/dt=1/(n+1)\) in differential form, which suggests a constant term associated with the \(\tau\)-equation.  
When strain is applied to the flow, all turbulence quantities are modified. In particular, the turbulence time scale \(\tau\) is pushed away from equilibrium. After the strain is removed, the turbulence tends to return to an equilibrium state. The modification adopted by [PERSON] et al. [5] to accomplish this is:  
\[\frac{d\tau}{dt}=\frac{1}{n+1}+C_{5}(z-z_{0}) \tag{3}\]  
where \(z=\varepsilon\tau/k\) and \(z_{0}\) is the value of \(z\) in isotropic decay flow (i.e., since the energy spectrum shape is preserved during decay, \(z_{0}=\left(\varepsilon\tau/k\right)_{0}\) is a constant, which gives \(\tau=z_{0}t/n\)). By using the \(\varepsilon\)-transport equation, \(z_{0}=n/(n+1)\) can be deduced, which shows that \(z_{0}\) is the reciprocal of \(C_{\varepsilon 2}\) in the \(k-\varepsilon\) model. [PERSON] et al. [5] chose \(z_{0}\) to be 0.54.  
By manipulating Equations (1)-(3), the following expression for \(z\) can be obtained:  
\[\frac{dz}{dt}=\frac{\varepsilon}{k}(1+C_{5})(z-z_{0}) \tag{4}\]  
In the (\(z\), \(dz/dt\)) phase plane, \((z_{0},0)\) is an equilibrium point. For this to be a stable point--in other words, to assure return to equilibrium--\(1+C_{5}\) must be negative or \(C_{5}<-1\).  
The final form of the model equations of the WFC model for homogeneous incompressible axisymmetric expansion flow is then given as (here, the WFC model is expressed in a form more suitable for the context):  
\[\frac{dk}{dt}=P_{k}-\varepsilon \tag{5}\]  
\[\frac{d\varepsilon}{dt}=-\frac{\varepsilon}{\tau}+C_{1}\frac{P_{k}\varepsilon }{k} \tag{6}\]\[\frac{d\tau}{dt}=\frac{1}{n+1}+C_{5}(z-z_{0})+C_{6}S\tau \tag{7}\]  
where \(P_{k}\) is the production of TKE and \(S\) is the mean strain rate. On the basis that \"production of dissipation\" estimated by the \(k\)-\(\varepsilon\) model is too weak (i.e., \(\varepsilon\) is under-predicted in high strain rate flows), [PERSON] et al. [5] tended to increase C\({}_{1}\) and tuned the coefficients to \(C_{1}=2\) and \(C_{6}=-2\). However, the choice of \(C_{1}=2\) does not satisfy spreading rate predictions for shear flows, as this coefficient tuned here for a plane jet is found to be \(C_{1}=1.44\).