## 3 Development of Viscoelastic Closures
### Closure for \(NLT_{ij}\)
The closure of Equation (26) is modelled in three parts: parts \(I\) and \(II\) are modeled in the same fashion as the model of [PERSON] et al. [32], part \(III\) is greatly improved and is the main contribution to the \(NLT_{ij}\) closure. Part \(I\) is approached by introducing the [PERSON]'s longitudinal micro-scale, \(\lambda_{f}\), to the relationship between the double correlation of fluctuating strain rates and the turbulent kinetic energy in homogeneous isotropic turbulence. More details can be found in [PERSON] et al. [32], with adjustments \(L^{0.42}\) to \(\sqrt{L}\) and \(f(C_{mm})^{0.8}\) to \(f(C_{mm})\). Part \(II\) is primarily responsible for capturing the shear component, \(NLT_{xy}\). The correlation here is with the exact term, \(M_{ij}\) (see Equation (13)), and by the local eddy viscosity, \(f_{N}^{1/4}\). The \(L^{0.15}\) variation is removed from the model developed by [PERSON] et al. [32]. The negative part of the \(NLT_{xx}\) component is also captured here via the \(M_{xx}\) term, which according to [PERSON] et al. [27], is the region where polymers inject energy into turbulence. Part \(III\) is developed to predict the \(NLT_{xx}\) component which is the dominant term in the trace of \(NLT_{ij}\), responsible for the stretch of the polymer chains due to turbulent fluctuations. Following the same assumption as [PERSON] et al. [29], one can see that \(NLT_{xx}\sim\overline{u_{x}^{\prime}u_{x}^{\prime}}\sim k\). In physical terms, the turbulent stretching terms represent the ability of the turbulent fluctuations to act on the polymer chains. This stretching is effective if the polymer shear and maximum extensibility are large enough. So, \(\sqrt{LM_{nn}/\dot{\gamma}}\) is included here with \(k\). Note that for fully developed channel flow, this term reduces to \(\sqrt{LC_{xy}}\) which increases proportionally to drag reduction. This new term includes the same physical assumption as [PERSON] et al. [28; 30], and is simplified from the very complex ad-hoc approach of [PERSON] et al. [32], viz  
\[NLT_{III}^{\text{Resende}}= f_{N}^{0.9}\exp\left(-\frac{-f_{N}}{1.05\sqrt{\beta}(10+0.3L+L-(L-1) ^{2})}\right)\] \[\times\left(\frac{C_{mm}}{(\beta/0.9)^{0.7\beta}}\left(2-\left[1- \exp\left(-\frac{2U_{b}h/v_{s}}{3500}\right)\right]^{4}\right)\right)^{0.7} \frac{\frac{dU_{i}}{dx_{k}}\frac{dU_{i}}{dx_{k}}}{\sqrt{\frac{dU_{p}}{dx_{q}} \frac{dU_{i}}{dx_{q}}}}. \tag{27}\]  
The performance of the \(NLT_{ij}\) closure can be analysed in Figure 1 by comparing the predictions with DNS data case (19) in Table 1, and with the model of [PERSON] et al. [32]. Figure 1a-c plots each normal component of \(NLT_{ij}\), with the predictions as accurate as the previous model [32]. The new \(NLT_{xx}\) component is capable of predicting the maximum value and peak location of the destruction effect away from the wall along with the negative part near the wall, but requires a much simpler closure. The closure performance becomes more noticeable at higher Reynolds numbers, in which the polymer extension is largely overestimated previously. The \(NLT_{yy}\) component is the leading order term in the \(C_{yy}\) component away from the wall, which is the dominant contributor to an effective polymeric viscosity. This strongly influences the turbulent dynamics according to [PERSON] et al. [40] and [PERSON] et al. [41] with their DNS and toy model analysis respectively. This term is represented by the first term in Equation (26), along with \(NLT_{zz}\). The \(NLT_{zz}\) component was shown by [PERSON] et al. [20] to have low impact, and thus \(NLT_{zz}=NLT_{yy}\) is an appropriate approximation. The shear component, \(NLT_{xy}\), can be viewed in Figure 1e, where the predictions omit similar results compared with the previous model [32], but do not require additional \(L^{2}\) variation via \(L^{0.15}\). Overall, all main features of \(NLT_{ij}\) are well captured such as the peak locations and magnitudes, but with a much simpler closure for the dominant contributor of polymer stretch, \(NLT_{xx}\).