*3.2. Changes in Drag, Lift, and Shedding Frequency*

We now turn to discuss how the relevant non-dimensional coefficients change as leading-edge undulation is applied. A summary of these quantities is given in Table 5, featuring the non-dimensional shedding frequency or Strouhal number *St*, lift coefficient RMS values *ClRMS*, the mean overall drag coefficient *Cd*, and also three sectional (2D) drag coefficients. The latter represent the drag of sections containing a valley, a peak, and those in the middle (for which the diameter matches the original, baseline one). The mean values for the overall drag coefficient and the RMS values of the lift coefficient were obtained from the time histories of these two quantities, which can be seen in Figure 7.

The main effects of leading-edge waviness gathered from Table 5 and Figure 7 are an increase in *Cd* and *ClRMS*, especially for case A11, and a slight decrease in the primary shedding frequency value *St*. We note that the latter have been evaluated from PSD plots of the lift fluctuation history, which are shown in Figure 8. It is also noteworthy that *St* values decrease gradually as the undulation amplitude is made larger, whereas the increase in *Cd* and *ClRMS* is more sudden, rising significantly from A03 to A11. This is related to a marked reduction in the base pressure coefficient *Cpb*, as will be discussed below in connection to Figure 10.

As made clear in the introduction, the typical effect of undulation in traditional wavy cylinders is to decrease the mean *Cd*. In order to help unveil why the opposite is being observed in the case of leading-edge waviness, the surface distributions of friction coefficient *Cf* and pressure coefficient *Cp* is discussed below. These distributions are shown in Figures 9 and 10, respectively.

**Figure 10.** Sectional mean *Cp* distributions for A03 (**a**) and A11 (**b**). A dashed curve denotes the baseline *Cp*.

The *Cf* distributions in Figure 9 show that the curves representing middle sections are not much different from the reference *Cf* curve of the baseline cylinder (dashed). For the other sections (peak and valley), each *Cf* curve changes in such a symmetrical way that the overall friction drag for those sections is not significantly altered. This indicates that the changes in overall *Cd* due to undulation are primarily caused by changes in pressure distribution. Still, important information that can be extracted from *Cf* plots has to do with flow detachment. The mean separation location can be defined from the point in which the *Cf* curve crosses zero (reversal of velocity profile near the surface). Moreover, the inclination (steepness) of the curve at this point of crossing typically indicates how abrupt the separation is. Hence, by looking at Figure 9, one can conclude that separation is anticipated at valleys and delayed at peaks, these effects being stronger at larger undulation amplitudes. Moreover, it seems that separation at valleys can be rather abrupt (especially for case A11), whereas separation at peaks seems to be much smoother. This is in fact the case, as will be discussed in connection to the iso-surfaces of mean streamwise velocity at near zero value, to be shown later, where vigorous boundary layer detachments are observed when separation occurs at the valleys.

Now, turning to the *Cp* distributions of Figure 10, one can see why case A03 has no significant change in *Cd* when compared to the baseline cylinder, namely, there is no significant change in pressure distribution, even along the spanwise direction. However, when it comes to case A11, the situation changes dramatically, especially with regard to the base pressure (*Cpb*) which becomes significantly reduced. This is likely the primary cause for the significant rise in mean *Cd* for case A11. For some reason, the base pressure seems to follow the pressure at the first location of separation (valley section), whose detachment occurs slightly before *θ* = 80 degrees, see Figure 9. Curiously, this does not happen with case A03, since a pressure recovery occurs from its first separation location until *θ* ≈ 100 degrees, after which the base pressure remains constant. It seems that for case A11, something takes between 80 < *θ* < 100 that prevents this pressure recovery. This will be further investigated in the next section through the analysis of the coherent structures that form in connection to the separation along valleys.

Lastly, it is worth mentioning that Figure 10 explains why the sectional drag at valleys is smaller than that at peaks, see Table 5. Essentially, undulation decreases the minimum pressure value on the *Cp* curve of valleys, imparting a suction on the frontal part of the section and therefore reducing its drag. The opposite effect is observed for the *Cp*

distribution of the peak section, which then contributes the most to the increased overall mean drag *Cd*. It is believed that the *Cp* reduction along valleys is caused by a Venturi-like effect due to the constriction of fluid between to peaks.
