## 3 Development of Viscoelastic Closures
### Model for the Modified Damping Function, \(f_{\
u}\)  
There have been many attempts to predict the eddy viscosity reduction as flow viscoelasticity increases for drag-reducing flows. In the case of low-Reynolds \(k-\varepsilon\) models for FENE-P fluids, this was examined firstly by [PERSON] et al. [20] for the LDR regime; then later by [PERSON] et al. [24] for the IDR regime. In both cases, there was a consistent reduction in the magnitude of \(k\) as DR increased, contrary to the DNS findings [23]. Similar attempts to model a modified damping function were made by [PERSON] [15]; [PERSON] et al. [16]; [PERSON] et al. [17] and [PERSON] [31] to develop viscoelastic turbulence models using different constitutive equations. Recently, [PERSON] et al. [32] proposed a modified damping function which was able to predict the correct behavior of the eddy viscosity close to the wall, leading to the appropriate increase for the magnitude of \(k\), and the shift away from the wall into the buffer layer as DR increased. This proposal was founded from the a-priori DNS data analysis by [PERSON] et al. [42], demonstrating the necessary increase to the production of \(k\) close to the wall. The model derived by [PERSON] et al. [32] is based on the DNS analysis of [PERSON] et al. [23], with an approximation of the form \(DR\sim C_{kk}/L\), giving rise to the correct damping of near-wall eddies as DR increases. In the \(k-\varepsilon-v^{2}-f\) models proposed by [PERSON] et al. [26] and [PERSON] et al. [28], the near-wall eddy viscosity damping effect is achieved by \(v^{2}\), as \(v_{T}=C_{\mu}v^{2}k/\varepsilon\). However, the reduction in \(v^{2}\) is not enough to increase \(k\) as given by the DNS data. The approach by [PERSON] et al. [32] works well in increasing \(k\) in the buffer layer, but fails to capture the viscoelastic effects away from the wall, due to the fact that \(f_{\mu}^{\rm Previous}\to 1\) as \(y\to h\), which is contrary to the DNS data of [PERSON] et al. [22] and the analogous behavior of \(v^{2}\) away from the wall. Therefore an additional model is required to capture the effect of nonequilibrium away from the wall, similarly to the Newtonian model of [PERSON] et al. [43]. [PERSON] et al. [41] demonstrated that the overall effect of polymer stretching is to introduce an effective viscosity proportional to \(C_{yy}\), which is dominated by the \(NLT_{yy}\) component (modeled here with the first term in Equation (26)). An additional term is multiplied to the eddy viscosity to account for the global reduction of eddy structures for increasing DR. This approach is similar to the model of [PERSON] et al. [24] and the study using DNS data of [PERSON] et al. [42] which multiplies the damping function by a factor of \(1-g(\rm VE)\), where \(g(\rm VE)\) is a function of the viscoelastic terms, VE. The final model presented for the modified damping function, \(f_{\
u}\), is  
\[f_{\
u}=(1-A)\left[1-\exp\left(-\frac{y^{*}}{a_{\mu}\left(1+B/a_{\mu}\right)} \right)\right]^{2}, \tag{28}\]  
\[A=C_{A}\left(f_{N}\frac{\lambda^{2}L^{3/2}}{f(C_{kk})^{2}}\frac{\varepsilon}{ v_{0}}\right)^{0.3}, \tag{29}\]  
\[B=C_{B}(C_{kk}-3)^{1.25}/L, \tag{30}\]  
with model constants \(C_{A}=0.071\) and \(C_{B}=0.44\). An additional contribution in the present model comes from an alternative representation of the dimensionless wall scaling \(y^{+}=u_{\tau}/v_{w}\), where \(v_{w}\) is the wall viscosity. The presence of the wall friction velocity poses a problem for flows with re-circulation or reattachment were the friction velocity becomes null at these points, causing floating point errors within computational solvers. Possibilities other than \(y^{+}\) that solve this issue are \(Re_{y}\equiv ky/v_{0}\) or the turbulent Reynolds number, \(Re_{t}\). [PERSON] and [PERSON] [44] formulated an alternative scaling, \(y^{*}\), in terms of \(Re_{y}\) so that \(y^{*}\approx y^{+}\) for \(y^{+}\leq 100\) in channel flows. The form proposed is  
\[y^{*}=C_{y1}Re_{y}^{1/2}+C_{y2}Re_{y}^{2}, \tag{31}\]where \(C_{y1}=2.4\) and \(C_{y2}=0.003\).