*2.5. Mesh Dependence Study*

A grid independence study was performed with four different mesh refinements for the smooth cylinder case, by evaluating the convergence of some reference variables (*Cd* and *CRMS*). Without loss of generality, the resolution was varied only via h-type refinement, the p-type refinement is the same in all cases, *Np* = 2. Therefore, the total DOF can be calculated by multiplying *NTotal* by 23. The results in Table 3 exhibited a suitable convergence of such variables, with little difference between the values of the medium grid, currently selected for this work, and the fine grid. After confronting it with experimental and computational results from the literature, as shown in the results section, the selection of the medium grid was considered adequate, and the same resolution was assigned to the wavy cases since the parameters of the wavy geometric modification are relatively small and produce a geometry smoothly close to the straight cylinder.

**Table 3.** Grid independence study: summary of tested cases with different mesh resolutions. The p-type refinement is the same in all cases, *Np* = 2. Relative to h-type refinement, *Nx*−*<sup>y</sup>* is the number of quadrilateral elements in *x*–*y* plane, *Nz* is the number of divisions in spanwise direction, and *NTotal* is the total number of hexahedral elements.


#### **3. Results and Discussion**

*3.1. Numerical Validation*

First of all, the results for the straight model, which is the baseline for this study, were compared to those available in the literature in Table 4. We note that our implicit LES results are closer to the DNS of Beaudan and Moin [50] than to the classic LES of Kravchenko and Moin [51]. The good agreement of these main flow variables serves as a primary validation for the numerical methodology (including the mesh) employed here. Values of lift and drag coefficients over time for the straight case are shown in Figure 7a. The Strouhal number was extracted from the main peak value of the PSD curve, see Figure 8a, calculated from the temporal history of *Cl* (Figure 7a). The separation angle *θ<sup>s</sup>* was estimated considering the criterion of *Cf* = 0 applied to the curve of skin friction coefficient of the baseline model given in Figure 9.


**Table 4.** Comparison and validation of results for the straight cylinder at Reynolds 3900.

**Figure 8.** Power spectral density of lift fluctuations for the baseline (**a**) and wavy cases A03 (**b**) and A11 (**c**).

**Figure 9.** Sectional mean *Cf* distributions for A03 (**a**) and A11 (**b**). A dashed curve denotes the baseline *Cf* .

In addition, the *ClRMS* value, see Table 5, is in the range of typical values found for this Reynolds number, which is around 0.08 for *Re* = 3900 [56]. Nevertheless, this Reynolds number is still reasonably close to the range in which the so-called lift crisis takes place (260 < *Re* < 1600), where *ClRMS* has a large behavioral variation, in which *ClRMS* values drop dramatically from ≈0.5 to 0.045, that is, a more significant variation could be observed. Notwithstanding, as shown in Table 4, all mean values of *Cd* and *ClRMS* are in line with experimental and computational results, as well as the Strouhal number, the separation angle and the base pressure coefficient.


A11 0.201 0.245 1.212 1.068 1.316 1.184

**Table 5.** Summary of relevant non-dimensional coefficients for the simulated cases.
