###### Abstract
The model can predict all DR regimes but is generally unattractive due to the higher number of Newtonian terms resulting from higher-order modeling. [PERSON] et al. [30] then further improved the \(k-\varepsilon-\overline{v^{2}}-f\) model capabilities via the \(NLT_{ij}\) term by introducing a simple extension to include heat transfer, along with removing wall dependence via the friction velocity. There are concerning features when one examines the Boussinesq-type \(NLT_{ij}\) term, which has a zero \(NLT_{yy}\) component, along with an opposite sign for \(NLT_{xy}\), both terms being crucial for the polymer shear stress in the momentum balance (see Appendix 1 in [PERSON] et al. [20]). Further, the increase of \(k\) in the buffer layer is small, meaning the decoupling of the \(v^{2}\) component may not be enough to decrease the eddy viscosity. This is compensated by an opposite trend in the dissipation rate, \(\varepsilon\), for increasing DR, which subsequently balances the momentum equations and causes the necessary increase in the velocity profiles. An alternative approach in predicting DR flow features other than the use of higher-order models such as the \(k-\varepsilon-v^{2}-f\) and Reynolds stress models mentioned previously, is that of a modified damping function or polymer eddy viscosity, accounting for the effect the polymer has on reducing the Reynolds shear stress. [PERSON] and [PERSON] [31] proposed a modified damping function for a low-Reynolds \(k-\varepsilon\) model for fluids described by the Giesekus constitutive equation, following the same ideas as [PERSON] [15] and [PERSON]. The closure was developed based on the energy-dissipative range and the dynamic characterization of the viscoelastic fluid. The model successfully captures the increase in the magnitude of the turbulent kinetic energy, along with the shift through the buffer layer. Although the magnitudes of \(k\) are largely over-predicted in many cases, which is counterbalanced by a lack of closure for the viscoelastic destruction term. In some instances, the model predicts a DR of 1% with a DNS result of 23%. [PERSON] et al. [32] proposed a modified damping function for a low-Reynolds number \(k-\varepsilon\) model for FENE-P fluids which can capture the increase of turbulent kinetic energy as flow viscoelasticity is increased, improving on model predictions made previously by [PERSON] et al. [20] and [PERSON] et al. [24]. The study also improved largely on the \(NLT_{ij}\) closure accuracy and simplicity formulated in the previous work. The model is able to predict flow features for a large range of rheological parameters but is limited to a friction Reynolds number of \(Re_{\tau_{0}}=395\), along with friction velocity still present in the model. For model applicability in flows with reattachment, the friction velocity dependence poses a problem as the values become null at these points and lead to spurious results or floating point errors within computational solvers. In the present study, an improved \(k-\varepsilon\) model for FENE-P fluids is proposed, validated for all drag reduction regimes (low, intermediate, high) and up to the largest friction Reynolds number (\(Re_{\tau_{0}}=1000\)) available in the DNS data. The important contribution to the current model is improved and simplified \(NLT_{ij}\) term that removes complexity from the most recent model developed by [PERSON] et al. [32]; along with a modified damping function which accurately predicts the viscoelasticcontributions near and away from the wall, effectively reducing the eddy viscosity and thickening the buffer-layer as DR increases. Further, a reformulation of the viscoelastic destruction term, \(E_{\tau_{p}}\), which removes all friction velocity present in the previous \(k-\varepsilon\) models.