## 2 Governing Equations  
The governing equations for incompressible turbulent flow of dilute polymer solutions are the continuity and momentum equations respectively:  
\[\frac{\partial\dot{u}_{k}}{\partial x_{k}}=0, \tag{1}\]  
\[\rho\frac{\mathrm{D}\dot{u}_{i}}{\mathrm{D}t}\equiv\rho\left(\frac{\partial \dot{u}_{i}}{\partial t}+\dot{u}_{k}\frac{\partial\dot{u}_{i}}{\partial x_{k} }\right)=-\frac{\partial\dot{\rho}}{\partial x_{i}}+\frac{\partial\dot{u}_{k} }{\partial x_{k}}, \tag{2}\]  
where the hat represents instantaneous quantities of velocity \(\dot{u}_{i}\), pressure \(\hat{p}\), stress tensor \(\hat{\tau}_{ij}\), and fluid density \(\rho\). The stress tensor is the sum of the Newtonian solvent which obeys [PERSON]'s law of viscosity, \(\hat{\tau}_{ij}^{s}=2\mu_{s}\hat{s}_{ij}\), with \(\mu_{s}\) representing the solvent viscosity coefficient, and polymeric contributions, \(\hat{\tau}_{ik}^{p}\),  
\[\hat{\tau}_{ij}=\hat{\tau}_{ij}^{s}+\hat{\tau}_{ij}^{p}. \tag{3}\]  
The kinematic viscosity is used alternatively throughout this study and is defined as \(\
u=\mu/\rho\). The instantaneous rate of strain tensor, \(\hat{s}_{ij}\), is defined as  
\[\hat{s}_{ij}=\frac{1}{2}\left(\frac{\partial\dot{u}_{i}}{\partial x_{j}}+ \frac{\partial\dot{u}_{j}}{\partial x_{i}}\right). \tag{4}\]  
The instantaneous polymer contributions are based on the FENE-P rheological dumbbell model [33], with closure given by  
\[\hat{\tau}_{ij}^{p}=\frac{\mu_{p}}{\lambda}\left(f(\hat{e}_{kk})\hat{e}_{ij}- \hat{s}_{ij}\right), \tag{5}\]  
with  
\[f(\hat{e}_{kk})=\frac{L^{2}-3}{L^{2}-\hat{e}_{kk}}, \tag{6}\]  
known as the Peterlin function, and \(\hat{e}_{kk}\) is the trace of the instantaneous conformation tensor. The other parameters that are associated with the FENE-P model are: \(\lambda\), the relaxation time of the polymeric fluid; \(L^{2}\), the maximum extensibility of the dumbbell model; and \(\mu_{p}\), the polymer viscosity coefficient. The behaviour of the instantaneous conformation tensor follows a hyperbolic differential equation of the form,  
\[\frac{\partial\hat{e}_{ij}}{\partial t}+\dot{u}_{k}\frac{\partial\hat{e}_{ij }}{\partial x_{k}}-\left(\hat{e}_{kj}\frac{\partial\dot{u}_{i}}{\partial x_{ k}}+\hat{e}_{ik}\frac{\partial\dot{u}_{j}}{\partial x_{k}}\right)=\frac{\
abla}{ \hat{e}_{ij}}=-\frac{\hat{\tau}_{ij}^{p}}{\mu_{p}}. \tag{7}\]The Oldroyd's upper convective derivative of the instantaneous conformation tensor is here denoted with \(\vec{\hat{c}}_{ij}\). The local and advective derivatives are the first and second terms respectively. The bracketed term accounts for the effect of polymer stretching by the instantaneous flow. The Reynolds averaging process [34] is applied to the governing equations via a Reynolds decomposition of the flow fields such that, \(\dot{u}_{i}=U_{i}+u_{i}\); where the use of overbars or upper-case represents the averaged quantity; and primes or lower-case represent the instantaneous quantities. The continuity and momentum equations now take the form:  
\[\frac{\partial U_{k}}{\partial x_{k}}=0, \tag{8}\]  
\[\rho\frac{\partial U_{i}}{\partial t}+\rho U_{k}\frac{\partial U_{i}}{ \partial x_{k}}=-\frac{\partial\overline{p}}{\partial x_{i}}+\mu_{s}\frac{ \partial^{2}U_{i}}{\partial x_{k}\partial x_{k}}-\frac{\partial}{\partial x_{ k}}(\rho\overline{u_{i}u_{k}})+\frac{\partial\overline{r}_{ik}^{p}}{\partial x _{k}}, \tag{9}\]  
referred to as the Reynolds-averaged Navier-Stokes (RANS). The Reynolds stress tensor is \(\overline{u_{i}u_{k}}\) and requires a closure model. The Reynolds-averaged polymer stress is \(\overline{r}_{ik}^{p}\) and written fully as  
\[\overline{r}_{ij}^{p}=\frac{\mu_{p}}{\lambda}\left[f(C_{kk})C_{ij}-\delta_{ ij}\right]+\frac{\mu_{p}}{\lambda}\overline{f(C_{kk}+c_{kk})c_{ij}}, \tag{10}\]  
where the additional term on the right requires a closure.