**4. Conclusions**

We presented a numerical study on the flow past cylinders with leading-edge waviness. This type of waviness resembles the tubercles found on the flippers of humpback whales and has been often studied in the context of wavy wings. To the authors' knowledge, this was the first study to consider this type of waviness for cylinders, since traditional studies on wavy cylinders typically apply undulation both to leading and trailing edges.

Some of the results obtained here were similar to those of traditional wavy cylinders, such as anticipated separation at valleys (sections with smaller diameters) and delayed at peaks (sections with larger diameters). Still, most flow changes affected by leading-edge undulation were found to be in contrast to those observed in traditional wavy cylinders. An explanation for these unexpected effects was then proposed based on a detailed flow field analysis. Specifically, the formation of counter−rotating streamwise coherent vortices was correlated to the delay of separation along peaks. Those vortices seem to push outer fluid toward the surface prior to separation at peaks, increasing overall velocity and reducing the pressure right before separation, especially for case A11. This defined a significantly lower base pressure (up to about 36%) and caused the observed rise in drag (up to about 28%). This decrease in base pressure likely draws the near wake features closer to the cylinder, for example, by reducing the recirculation length. Since the turbulent structures of the near wake become more "compressed" closer to the cylinder—compared to the straight cylinder, the turbulent kinetic energy rises on the near wake and the larger oscillations in *y*-velocity of the vortex formation zone impart larger oscillations in the lift. As all these combined effects are far from simple, subsequent studies could further clarify the underlying dynamics.

It seems that the most relevant aspect in the reduction of base pressure is the strength of the coherent streamwise vortices. Their strength is nevertheless related to undulation (wavelength and amplitude), as well as to the Reynolds number. It is worth mentioning that an experimental study currently being conducted on similar geometries, see [13], but at much larger Reynolds numbers, showed a reduction in drag instead, which confirms its Reynolds number dependence.

Although the results did not replicate the feats of the wavy airfoil or even the typical wavy cylinders, this study was able to offer a good phenomenological description that may be useful in understanding and complementing these other studies. Furthermore, a practical application for it beyond flow control mechanisms, by taking advantage of the observed mixing layer effect promoted by the streamwise vortices, would be in circular tubes of heat exchangers. Additionally, more studies are being conducted to explore the possibilities of this geometric modification to different parameters combinations.

**Author Contributions:** Conceptualization, P.H.F., T.B.d.A. and R.C.M.; methodology, P.H.F. and R.C.M.; software, P.H.F., E.O.C. and L.D.F.; validation, P.H.F. and R.C.M.; formal analysis, P.H.F., T.B.d.A., E.O.C., L.D.F. and R.C.M.; investigation, P.H.F., T.B.d.A. and R.C.M.; data curation, P.H.F.; writing—original draft preparation, P.H.F. and R.C.M.; writing—review and editing, P.H.F.; visualization, P.H.F., E.O.C. and L.D.F.; supervision, R.C.M. and T.B.d.A.; funding acquisition, R.C.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by FAPESP grant number 2020/10910-8. The APC was funded by CNPq (process number 141783/2019-2).

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank André F. C. da Silva and Mohsen Lahooti for their support regarding the efficient use of Nektar++ in the context of parallel computations. In addition, the first author acknowledges support from CNPq for his Ph.D. scholarship (process number 141783/2019-2).

**Conflicts of Interest:** The authors declare no conflict of interest.
