## 3 Development of Viscoelastic Closures
### Closure for \(NLT_{ij}\)  
The \(NLT_{ij}\) exact transport equation is greatly simplified based on the DNS analysis of [PERSON] et al. [20]: Following the transport equation of \(\overline{f(\hat{c}_{mm})c_{kj}\frac{\partial u_{i}}{\partial x_{k}}}+\overline{ f(\hat{c}_{mm})c_{ik}\frac{\partial u_{j}}{\partial x_{k}}}\), it is assumed that  
\[\overline{f(\hat{c}_{mm})c_{kj}\frac{\partial u_{i}}{\partial x_{k}}}+\overline {f(\hat{c}_{mm})c_{ik}\frac{\partial u_{j}}{\partial x_{k}}}\approx f(C_{mm}) \left(\overline{c_{kj}\frac{\partial u_{i}}{\partial x_{k}}}+\overline{c_{ik} \frac{\partial u_{j}}{\partial x_{k}}}\right)=f(C_{mm})NLT_{ij}. \tag{25}\]  
The full details of this approximation and the exact transport equation of \(NLT_{ij}\) can be found in [PERSON] et al. [20] and [PERSON] et al. [24]. The complete closure of \(NLT_{ij}\) is presented below and was developed to improve model predictions based on better physical modeling compared with the most recent model developed by [PERSON] et al. [32]. \[NLT_{ij} =\overline{c_{kj}\frac{\partial u_{i}}{\partial x_{k}}}+\overline{ c_{ik}\frac{\partial u_{j}}{\partial x_{k}}} \tag{26}\] \[\approx\underbrace{f_{N}C_{N1}\frac{\lambda\sqrt{L}e^{N}}{v_{0}f (C_{mm})}\delta_{ij}}_{I}-\underbrace{f_{N}^{1/4}C_{N2}M_{ij}}_{II}+ \underbrace{C_{N3}\frac{k}{v_{0}}\sqrt{\frac{\bar{L}M_{m}}{\gamma}\frac{ \partial U_{i}}{\partial x_{k}}\frac{\partial U_{j}}{\gamma}}}_{III},\]  
where \(f_{N}=v_{T}/v_{0}\) is the local eddy viscosity, \(\dot{\gamma}=\sqrt{2S_{pq}S_{pq}}\) is the shear rate invariant, \(\bar{L}=\sqrt{L^{2}/900}\) is the normalised maximum extension with the lowest DR, with model constants \(C_{N1}=0.11\), \(C_{N2}=0.3\) and \(C_{N3}=0.3\). \begin{table}
\begin{tabular}{c l c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Reference**} & \multicolumn{3}{c}{**Rheological Parameters**} & \multicolumn{3}{c}{**Drag Reduction (\%)**} \\ \cline{3-8}  & & \(Re_{\tau_{b}}\) & \(Wi_{\tau_{b}}\) & \(L^{2}\) & **DNS** & **Current Model** & **Model [32]** \\ \hline (1) & [PERSON] et al. [23] & 125 & 25 & 900 & 19 & 20 & - \\ (2) & [PERSON] et al. [23] & 125 & 25 & 3600 & 22 & 23 & - \\ (3) & [PERSON] et al. [23] & 125 & 25 & 14,400 & 24 & 25 & - \\ (4) & [PERSON] et al. [23] & 125 & 50 & 900 & 31 & 30 & 35 \\ (5) & [PERSON] et al. [23] & 125 & 100 & 900 & 37 & 36 & 39 \\ (6) & [PERSON] et al. [23] & 125 & 100 & 1800 & 45 & 43 & - \\ (7) & [PERSON] et al. [23] & 125 & 100 & 3600 & 56 & 51 & 51 \\ (8) & [PERSON] et al. [28] & 180 & 25 & 900 & 19 & 19 & - \\ (9) & [PERSON] et al. [23] & 180 & 50 & 900 & 31 & 30 & 34 \\ (10) & [PERSON] et al. [28] & 180 & 100 & 900 & 38 & 38 & 39 \\ (11) & [PERSON] et al. [28] & 180 & 100 & 3600 & 54 & 53 & 51 \\ (12) & [PERSON] et al. [37] & 180 & 116 & 10,000 & 64 & 60 & - \\ (13) & [PERSON] et al. [26] & 300 & 36 & 3600 & 33 & 32 & 34 \\ (14) & [PERSON] et al. [26] & 300 & 36 & 10,000 & 35 & 35 & 32 \\ (15) & [PERSON] et al. [26] & 300 & 120 & 10,000 & 59 & 59 & 58 \\ (16) & [PERSON] et al. [30] & 395 & 25 & 900 & 19 & 22 & 19 \\ (17) & [PERSON] et al. [30] & 395 & 50 & 900 & 30 & 30 & - \\ (18) & [PERSON] et al. [30] & 395 & 50 & 3600 & 38 & 38 & - \\ (19) & [PERSON] et al. [30] & 395 & 100 & 900 & 37 & 37 & 38 \\ (20) & [PERSON] et al. [30] & 395 & 100 & 3600 & 48 & 47 & 52 \\ (21) & [PERSON] et al. [39] & 395 & 100 & 10,000 & 55 & 55 & - \\ (22) & [PERSON] et al. [30] & 395 & 100 & 14,400 & 61 & 60 & 62 \\ (23) & [PERSON] et al. [37] & 395 & 116 & 10,000 & 62 & 60 & - \\ (24) & [PERSON] et al. [23] & 395 & 200 & 14,400 & 75 & 69 & 67 \\ (25) & [PERSON] et al. [30] & 590 & 50 & 3600 & 39 & 40 & 64 \\ (26) & [PERSON] et al. [37] & 590 & 116 & 10,000 & 61 & 59 & 74 \\ (27) & [PERSON] et al. [38] & 1000 & 50 & 900 & 30 & 33 & 60 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Independent Direct Numerical Simulation data for turbulent channel flow of finitely extensible nonlinear elastic-Peterlin (FENE-P) fluids at \(\beta=0.9\), with drag reduction (DR) model predictions.