# The Role of Elasticity in the Vortex Formation in Polymeric Flow around a Sharp Bend
## 2 Materials and Methods
### Computational Methodology
#### 2.2.4 Meshing  
Figure 2 presents an enlarged portion of the bend corner geometry with the progression in mesh refinement used for the simulations. Preliminary runs were performed in the coarse mesh shown in Figure 2a. The mesh was halved twice for the final run of every simulation, Figure 2c. An even finer fourth mesh was used in a few simulations to estimate grid convergence regarding the size predicted for inertial and elastic vortices including the phase maps.  
Figure 3 presents a sample of the grid convergence analysis performed in choosing the grid for the simulations and estimate the numerical errors of the predictions. The profiles in the figure correspond to a case with an elastic vortex located upstream of the bend corner. They were created along the white vertical line highlighted in Figure 2c; the grid chosen for the numerical analysis. The line crosses the elastic vortex roughly in the middle.  
Figure 3a compares the streamwise velocity component for the four grids evaluated. The weighted average deviation error outside the vortex for the fine grid is 1.3% and 5.5%  
Figure 3: Grid convergence analysis, \(\text{Wi}=1.0\), \(\text{El}=100\), \(\beta=0.1\): (**a**) horizontal velocity; (**b**) vertical velocity; (**c**) axial stress.  
inside the vortex. Similarly, Figure 3b compares the spanwise velocity component for the same four grids. The weighted average deviation error outside the vortex is 14% and 26% inside the vortex, respectively (in both cases, note the very low actual value of the predicted velocity). Lastly, Figure 3c compares the axial stress. The weighted average deviation error outside the vortex is 4.0% and 8.3% inside the vortex, respectively.  
Similar analyses were conducted for many cases, including those with inertial vortices located downstream of the bend corner. In a typical high Reynolds number case, Re = 100, the weighted average deviation errors of all variables are significantly smaller. For the streamwise velocity component, the weighted average deviation error outside the vortex for the fine grid is 0.02% and 0.4% inside the vortex, respectively. For the spanwise velocity component, the weighted average deviation error outside the vortex is 0.09% and 0.5% inside the vortex, respectively. Additionally, for the strain rate, the weighted average deviation error outside the vortex is 0.04% and 0.28% inside the vortex, respectively.  
In general terms, it was found that predicting the location and length of the vortex is the most significant challenge in attaining grid convergence. Outside the vortex and far from it, the grid convergence index (GCI) for the fine grid is excellent, ranging from 0.0 to 1.5% in all variables. Within the vortex, the GCI for the fine grid is acceptable, ranging from 0.1% to 25% in all variables. Again, it should be noted that actual values for the spanwise velocity and the normal stress within the vortex are significantly smaller when compared to the streamwise velocity and the axial stresses, by two to three orders of magnitude. The mesh and grid convergence analyses were conducted following the Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications by [PERSON] et al. [23].  
The formation of vortices in polymer flow has been studied extensively in planar contractions and the size and strength have been shown to be strongly dependent on the mesh [15; 24; 25]. For example, the effect of mesh refinement on the size of the vortices was highlighted by [PERSON] et al. [26] when developing benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions. They found that a very high degree of mesh fineness was required to obtain accurate results with the Oldroyd-B fluid, while the PTT fluid in general did not require the finest meshes. These authors relied on [PERSON]'s extrapolation to measure the level of convergence.