*3.3. Flow Field Analysis and Coherent Structures*

We start by looking at two-dimensional plots of mean spanwise vorticity *ω<sup>z</sup>* for the cylinders' three relevant sections, as shown in Figure 11. Note that these plots show the correct size of each cylinder's section, with valleys having smaller diameters and peaks having larger ones — except for the baseline cylinder, whose plots in the top row of Figure 11 are simply repeated for convenience. An important effect of undulation, as discussed in the previous section, is to delay separation at peaks and to anticipate it at valleys. This effect is clearly seen in Figure 11, especially at the bottom row of plots (case A11). In the latter, one can see that the shear layers behind a peak section are closer to one another as they bracket the recirculation region (near wake). They also originate from the cylinder at a greater azimuthal coordinate (separation angle). In contrast, for the valley section, the shear layers detach from the cylinder much earlier, becoming farther away from one another along the recirculation zone. These will visually look like strong "ejection-like" events subsequently to be discussed later. These observations are in line with studies on traditional wavy cylinders, although flow detachment, even when anticipated at valleys, is typically not as strong as the ones found here.

**Figure 11.** Two−dimensional contours of mean spanwise vorticity *ω<sup>z</sup>* for the three relevant sections (see the header on top) of the simulated cylinders: baseline (**a**), A03 (**b**), and A11 (**c**). The baseline model plots are repeated on the top row.

Another effect observed in Figure 11 is that leading−edge undulation causes the recirculation region at the near wake to reduce its length in the streamwise direction. This effect, nevertheless, is in contrast to studies on traditional wavy cylinders, where the recirculation length typically becomes larger in comparison to that of the straight cylinder. In the literature, an increase in recirculation zone length is normally associated with a reduction in turbulent kinetic energy at the near wake. Here, we found that this trend is followed in the sense that the observed reduction in recirculation zone length leads to an increase in turbulent kinetic energy (TKE), as shown in Figure 12. This figure shows a carpet-like view in the *y* = 0 plane for the three cylinders, whose shapes (cross-sections) appear in white color.

For each cylinder, Figure 12 shows a relatively small TKE variation in the spanwise direction (note each plot repeats itself periodically along the span, following the undulation wavelengths), but it is clear that maximum TKE is reached behind valleys. However, when comparing the cylinders, it becomes clear that leading-edge undulation not only increases overall TKE levels in the near wake, but also brings the high-TKE zone closer to the cylinder, which is consistent with the reduction in recirculation length. Turbulence along the wake can also be assessed through the energy spectrum of the velocity components, as given in Figure 13, which also shows the pressure spectrum (all these are span-averaged).

**Figure 12.** Carpet−like view of mean turbulent kinetic energy in the *y* = 0 plane: baseline (**a**), A03 (**b**), and A11 (**c**). As a spanwise periodic phenomenon, the spatial average referring to the repetition of wavelengths is calculated and exhibited here as a single wavelength.

**Figure 13.** Energy spectrum of velocity components and of pressure at *y* = 0 and different *x* positions (see top header) for the three simulated cylinders: baseline (**a**), A03 (**b**), and A11 (**c**).

The positions along the wake where the spectra in Figure 13 have been measured can be correlated with the coordinates shown in Figures 11 and 12. Kolmogorov's −5/3 slope is included in the plots of Figure 13 for reference and its agreement with the measured spectra over the inertial range of the turbulence cascade further corroborates the suitability of the numerical methodology (including the mesh) employed. The rise in TKE intensity on the near wake due to undulation can be noted by comparing the spectra in the left column of Figure 13, as the curves move upwards from plots (a) to (c). Moreover, note that the peak in the *y* velocity component (*V* in the plots) increases especially for case A11 in the near wake, consistent with the rise in *ClRMS* discussed in the previous section, see Table 5. This also seems to be related to an anticipation of wake turbulence levels due to undulation, which

can be noted by comparing the spectra of case A11 at *x* = 1 with that of the baseline case at *x* = 3. In this comparison, note that not only the peak in *V* is anticipated, but also the peak in pressure. This indicates that the turbulent features of the wake are anticipated, which is in line with the recirculation length being reduced. As the more intense turbulent activity is then brought closer to the cylinder, it is not that surprising that stronger oscillations will affect the cylinder, which helps explain at least partially the rise in *ClRMS*. In summary, the strong oscillations in *y*-velocity that would otherwise take place further downstream are now happening at the near wake, affecting the vortex shedding itself in a way that increases oscillations in *Cl*. Lastly, it is worth noting that the spectra measured at the far wake (*x* = 9) do not change significantly among cylinders, as the turbulent wakes seem to relax toward a canonical turbulent wake state.

Finally, we turn to the analysis of streamwise vorticity, which will reveal the formation of strong (streamwise) vortex pairs associated with how separation is anticipated along valleys and delayed along peaks. In order to track the evolution of said vorticity, we first consider the frontal part of the cylinders by using the azimuthal coordinate *θ* as a parameter, while looking at the vorticity component *ω<sup>t</sup>* that is tangent to the (circular) cross-sections of the cylinder, as show in Figure 14 (left schematic). Hence, in Figure 15, *ω<sup>t</sup>* will be shown in the *r*–*θ* plane that contains the original (baseline) axis of the cylinders, for increasing values of *θ*. After *θ* = 90 degrees, i.e., for the rear part of the cylinders, the streamwise vorticity component *ω<sup>x</sup>* will be considered instead, as this will more naturally allow for the tracking of the (streamwise) vortex pairs that, at this point, are no longer attached to the cylinders' surface. Thus, in Figure 16, streamwise vorticity is shown over the *y*–*z* plane, with the *x* coordinate being used as the parameter.

**Figure 14.** Planes and coordinates chosen to best follow the evolution of vorticity along the surface and after separation: tangent component *ω<sup>t</sup>* for the cylinders' frontal part (**a**) and streamwise component *ωx* for the cylinders' rear part (**b**).

In Figure 15, *ω<sup>t</sup>* is shown for cases A03 (top) and A11 (bottom) for increasing values of *θ*. Note that *ω<sup>t</sup>* would be zero for the baseline cylinder (at least for the mean flow) since it represents a distributed vorticity field associated with the boundary layer flow that exists along the spanwise direction (cross-flow) due to surface undulation. For *θ* < 20 degrees, the plots basically indicate that, with respect to the spanwise direction, the boundary layer evolves from the peaks to the valleys, as expected near the leading-edge region. As *θ* increases, the direction of flow becomes reversed, as can be seen by how the colors have changed by *θ* = 80 degrees. This reversal is likely associated with flow separation, which first occurs at valleys around *θ* = 80 degrees, recall Figure 9. More specifically, the near-surface flow between peaks seems to be accommodating beforehand to bifurcate upon encountering the "ejections-like" by which separation at valleys occurs. These flow structures can be seen in Figure 17 and are surprisingly strong for case A11. Lastly, it is also worth noting in Figure 15 that the vorticity layer near the surface is thicker for case A11, which is likely caused by a stronger cross-flow due to larger undulation amplitude.

**Figure 15.** Contours of mean tangent vorticity *ωt* over the *r*–*θ* plane for the two wavy models, A03 (**a**) and A11 (**b**), for increasing values of *θ*. Refer to Figure 14 (left schematic) for the planes and coordinates adopted. As a spanwise periodic phenomenon, the spatial average referring to the repetition of wavelengths is calculated and exhibited here as a single wavelength.

In Figure 16, it becomes clear that each side of the vorticity layer will, upon flow separation at valleys, detach from the surface and, once free, will roll-up upon itself and become a coherent streamwise vortex. Therefore, the two sides of the vorticity layer between peaks will become a pair of counter-rotating vortices. Globally, the array of undulation wavelengths along the span generates a corresponding array of streamwise vortex pairs. Clearly, from Figure 16, case A11 features much stronger vortices than case A03, as expected from the overall amount of streamwise vorticity held in the respective vorticity layers prior to separation (recall comment at the end of the previous paragraph). The right-most pair of plots in Figure 16 correspond to the plane *x* = 0.5, which is the one containing the trailing edge of the cylinders. At this point, the vortices of case A03 are already vanishing, whereas those of case A11 are still strong and will surely extend further into the near wake region. This will be made more evident in Figure 18, in which streamlines are employed to highlight the coherent pair of streamwise vortices. In fact, as per Figure 18, the vortices of case A03 are not strong enough to produce a cyclical streamline swirl, indicating that only case A11 effectively produces coherent vortices.

**Figure 16.** Contours of mean streamwise vorticity *ωx* over the *y*–*z* plane for the wavy models, A03 (**a**) and A11 (**b**), for increasing values of *x*. Refer to Figure 14 (right schematic) for the planes and coordinates adopted. As a spanwise periodic phenomenon, the spatial average referring to the repetition of wavelengths is calculated and exhibited here as a single wavelength.

In Figure 17, iso-surfaces of mean streamwise velocity at a near zero value (*<sup>U</sup>* ≤ <sup>10</sup><sup>−</sup>3) are employed to highlight how the flow separation takes place for each cylinder. One can clearly see how separation first occurs at valleys by means of separated vortical shear layers of fluid moving away from the surface. These can be connected to the streamwise vortices by correlating Figure 17 to Figure 16. The vortex pair between two peaks rotates in such a way as to "lift" fluid from the surface at valleys and to "land" fluid on the surface at peaks. The former of these effects explains why such structures are stronger when vortices are stronger, whereas the latter explains why separation at peaks is further delayed when vortices are stronger, recall Figure 9.

**Figure 17.** Iso−surfaces of mean streamwise velocity at a near zero value (*<sup>U</sup>* <sup>≤</sup> <sup>10</sup>−3), showing how the flow separation at valleys occurs by means of "ejections−like" events. The three simulated cylinders are shown: baseline (**a**), A03 (**b**) and A11 (**c**).

In Figure 18, streamlines based on the mean flow are used to highlight the formation of the coherent vortices. They clearly show that only case A11 produces vortices that are effectively cyclical. The left-most figure of case A11 in Figure 18 shows how the vortex pair adjacent to a peak pushes outer fluid toward the surface. Since this happens prior to separation at peaks, the overall velocity increases in this region, and, therefore, pressure is reduced (Bernoulli equation). This is consistent with the *Cp* curve along peaks already shown (see Figure 10 for 80 < *θ* < 120 degrees). Hence, when separation occurs at peaks, the pressure at the surface is significantly reduced. This, combined with the fact that the earlier separation at valleys also happened at a location of reduced pressure (*θ<sup>s</sup>* ≈ 80 degrees), ends up allowing for reduced pressure on the separation zone and near wake. In summary, the mechanism proposed to explain the increased drag of case A11 due to low base pressure is that strong coherent vortices increase the overall velocity just before separation at peaks, leading to reduced pressure on the separation zone.

**Figure 18.** Mean flow-based streamlines showing the formation of coherent streamwise vortices for the wavy cylinders: A03 (**a**) and A11 (**b**). Only case A11 produces effectively complete cyclical vortices. The colors indicate local streamwise vorticity.
