**1. Introduction**

Many researchers have exhaustively studied flow over bluff bodies, especially cylinders, such as Roshko [1], Williamson [2], and Zdravkovich [3]. Whether due to the simplicity of the geometry or the intriguing complexity of the emerging flow structures, cylinders have always aroused this interest over time. Additionally, this geometry has always been present in everyday life, in different engineering applications, e.g., flow over buildings, wind turbines, and oil risers, which reinforces its importance, especially by knowing they suffer the action of a phenomenon called vortex-induced vibration (VIV). In a simplified way, this is an oscillating effect induced by the interaction of pressure fluctuations created by a vortex-shedding wake on the body itself, which can even compromise its structural integrity [4].

In the search for solutions to mitigate or suppress phenomena such as VIV, or simply to reduce drag or the mean lift fluctuations (i.e., RMS of lift forces), many studies have been carried out with passive flow control mechanisms. In other words, those mechanisms do not have an active device, but work by changing the body geometry itself, e.g., a surface

**Citation:** Ferreira, P.H.; de Araújo, T.B.; Carvalho, E.O.; Fernandes, L.D.; Moura, R.C. Numerical Investigation of Flow Past Bio-Inspired Wavy Leading-Edge Cylinders. *Energies* **2022**, *15*, 8993. https://doi.org/ 10.3390/en15238993

Academic Editors: Artur Bartosik and Dariusz Asendrych

Received: 3 October 2022 Accepted: 21 November 2022 Published: 28 November 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

undulation, like the one that will be presented here, or the addition of some fixed external structure to the body surface, such as the well-known vortex generator [5], that helps to improve stall characteristics.

Despite also having a circular cross-section, different from the planform of the typical existing wavy cylinders, the present work proposes a new bio-inspired planform cylinder, as shown in Figure 1, here named the "wavy leading-edge cylinder". The inspiration comes from works on wavy leading-edge wings, whose study was first motivated by the morphological observations made by Fish and Battle [6], in 1995, about humpback whales. This study hypothesized that the presence of some protuberances (also called tubercles) almost regularly distributed on the leading-edge surface of the humpback whale flippers acted like a flow control mechanism responsible for the distinct hydrodynamic characteristics of these animals, which are known to be excellent hunters, and are able to perform complex movements with great ease and under high angles of attack.

**Figure 1.** Types of wavy cylinders (in each cylinder, the leading edge appears on the left and the trailing edge on the right). The right-most cylinder shows the waviness adopted in the current work,

Different studies [7–12] mimicking those tubercles have tried to understand how this geometry allowed to improve maneuverability and whether they could enhance the aerodynamic performance of wings. Those studies have evidenced interesting phenomena and promising results, such as three-dimensional effects like the emergency of longitudinal counter-rotating vortex pairs (CVP), the formation of three-dimensional laminar separation bubbles (LSB), the delay of boundary layer detachment, a more evenly distributed surface pressure, an aerodynamic efficiency improvement (by reduction of drag and/or increase of lift), and so on.

whereas the other two cylinders show the typical patterns of waviness considered in the literature.

After this brief overview of wavy leading-edge airfoils, we turn to the theme of this work. The proposed waviness modification was created by giving volume to a geometry after sweeping a circular area through a planform constrained by a sinusoidal function curve at the leading-edge line and a straight line at the trailing-edge line, as shown in Figure 2. This is a computational part of a research that also includes an experimental part [13] for the same geometries, but at higher Reynolds numbers. As will be discussed in the present work, the applied waviness resulted in a drag increase at the current (lower) Reynolds of 3900, showing there is also an important Reynolds effect at play.

**Figure 2.** Sectional views—cutting plane (**top**) and three-dimensional view (**bottom**)—of the new flow control geometry proposed: the wavy leading-edge cylinder geometry. The illustrative figure presents the main parameters and constructive function in the *x*–*z* plane.

Here, simulations were conducted first to validate the numerical model for the straight cylinder at Re = 3900 and then to investigate two wavy geometries at the same Reynolds, namely two amplitudes for a single wavelength. At this Reynolds number, the straight cylinder is well within the subcritical regime, in which the boundary layer is laminar and transition occurs along the separating shear layer (via Kelvin–Helmholtz instability) leading to a turbulent wake featuring the well-known von-Kármán vortex street. In the near wake region, the interaction between the transitional shear layer and the wake creates threedimensional turbulent structures that evolve in both streamwise and spanwise directions.

Focusing on wavy cylinders, Ahmed and Bays-Muchmore [14] were among the first authors to carry out experiments with transverse flow over a wavy cylinder. In this case, they measured surface pressure in a wind tunnel for a Reynolds number of 10,000 and performed dye visualizations in a water tunnel for Reynolds numbers of 5000, 10,000, and 20,000. They tested a traditional sinusoidal wavy planform with symmetric waviness in the spanwise direction for a set of wavy models with wavelength (*λ*) ratio to the mean diameter *Dm* of *λ*/*Dm* = 1.2, 1.6, 2.0, and 2.4, and a fixed amplitude ratio of *A*/*Dm* = 0.1. They found that waviness reduces overall drag and observed that the sectional (2D) drag coefficients were greater at peaks (larger diameter section) than at valleys (smaller diameter section). Additionally, they found that separation was anticipated on valleys and delayed on peaks. Note that the frontal pressure typically changes along the spanwise direction, also influencing differences in drag between sections. In addition, the separation line maintained a curvature proportional to the wavy geometry applied, following a wavy pattern. Additionally, from flow topology analysis, they inferred the possible formation of a pair of longitudinal vortices in between adjacent peaks.

Zhang et al. [15] carried out experiments in a water channel at a Reynolds number of 3000 in order to investigate the near wake behavior of a wavy cylinder through a particle image velocimetry (PIV) measurement technique. From the transverse vorticity results, they inferred the formation of structures supposed to be a pair of counter-rotating streamwise vortices at both sides of each peak. These vortices showed an action to suppress the development of both large-scale spanwise vortices and regular vortex shedding. This effect indicated a possible reason for the reduction in the average turbulent kinetic energy (TKE) over the near-wake region, especially after the valley regions. Therefore, this consequential lower TKE also helped to justify lower values of drag coefficient compared to the straight cylinder.

Lam and Lin [16] conducted a large-eddy simulation (LES) to study the flow past wavy cylinders at a Reynolds number of 3000. They tested a combination of parameters for *λ*/*Dm* varying from 1.136 to 3.333, and two amplitude ratios, *A*/*Dm* = 0.091 and 0.152. The most notable result was a reduction of about 18% in drag when compared to the straight cylinder, for *λ*/*Dm* = 1.9 and *A*/*Dm* = 0.152. In addition, waviness caused a reduction or even a suppression of RMS lift coefficients with no significant changes in Strouhal number (*St*). Furthermore, they observed that waviness created a difficulty for the vortex sheet to roll up, which ended up moving downstream the vortex formation region. This three-dimensional free shear layer effect helped them to explain the less intense pressure fluctuations over the surface and the lower values of turbulent kinetic energy in the wake (compared to the "two-dimensional" wake of the straight cylinder), leading to higher base pressure values and lower drag coefficients.

Lin and Yu-fen [17] conducted a large-eddy simulation study, with an experimental validation of the straight model in a wind tunnel, to investigate the effect of waviness at a Reynolds number of 3000. They tested a single combination of wavelength and amplitude ratios, namely *λ*/*Dm* = 1.5 and *A*/*Dm* = 0.15. Their conclusions suggested that waviness imposed a difficulty for the formation of a vortex street, also with less intense pressure fluctuations and smaller values of turbulent kinetic energy over different spanwise crosssections in the wake. Those effects led to a reduction in both mean drag and RMS lift coefficients. Again, the separation was anticipated in the valleys and delayed in the peaks.

Lam et al. [18] performed a large-eddy simulation and an experimental validation with laser Doppler anemometry (LDA) and load cell measurements in water and wind tunnels, to investigate flow past a yawed wavy cylinder at a Reynolds number of 3900. They tested a single combination of wavelength and amplitude ratios, *λ*/*Dm* = 6.0 and *A*/*Dm* = 0.15, respectively, with yaw angle varying from 0◦ to 60◦. The results of the flow with yaw angle = 0◦ (or unyawed), which is the main interest for the present work, showed drag and RMS lift coefficients reductions of up to 14% and 80%, respectively. Power spectral density (PSD) revealed a lower *St*, namely 0.184 against 0.208 from the straight cylinder. Moreover, pressure and velocity distributions confirmed spanwise periodic and repetitive wake structures, following the pattern imposed by the wavy geometry.

Zhao et al. [19] implemented a scale-adaptive simulation to understand the physics of the flow past a wavy cylinder at Reynolds number 8000 for several configurations. They varied *λ*/*Dm* from 3 to 7 for *A*/*Dm* = 0.091 and 0.152. They confirmed results already obtained by other authors, with reductions in *Cd*, *St*, and *ClRMS* of up to 30%, 50%, and 92%, respectively, compared to the straight case. Besides associating the reduction of lift fluctuations to the presence of longitudinal vortices, they visualized for certain configurations a change in the pattern of surface streamlines after the separation line (curve). Lastly, they also observed longer mean recirculation lengths, which means spanwise vortex formation moved downstream, which led to a base pressure increase and corresponding drag reduction.

Bai et al. [20] investigated flow past wavy cylinders in a water tunnel at a Reynolds number of 3000 through a time-resolved PIV (TR-PIV) technique. They tested a single model of *λ*/*Dm* = 0.152 and *A*/*Dm* = 6. PSD results revealed two different vortex-shedding frequencies associated with peak and valleys sections, respectively, of *f* = 0.164 and

*f* = 0.214, the last one very close to the Strouhal number expected for a straight cylinder at this Reynolds number (*St* = 0.21), indicating that larger structures such as von-Kármán vortices still dominate flow. An analysis based on proper orthogonal decomposition (POD) in the spanwise direction corroborated this by showing that the first two dominant modes exhibit the spanwise vortex shedding in valleys and peaks and together concentrate almost 45% of the POD energy. Despite that, near the wake in the streamwise direction, the most energetic POD modes were associated with the presence of the streamwise vortices, which tend to suppress the coherence of spanwise vortices. In addition, they observed that mean recirculation lengths extend further downstream, more in peaks than in valleys, as well as when compared to the straight cylinder.

Karthik et al. [21] studied the flow past a wavy cylinder using LES along with the Ffowcs, Williams and Hawking's (FWH) acoustic analogy at Reynolds number 97,300. They tested all combinations of parameters for *λ*/*Dm* = 1.0, 1.5, 2.0, and 2.5, and *A*/*Dm* = 0.05, 0.1, 0.15, and 0.2. Drag and RMS lift reduced in all wavy cases, with the greatest drag reduction for *λ*/*Dm* = 2.0, and the minimum RMS lift for *λ*/*Dm* = 1.5, among all amplitudes. The parameters for the optimum case for both drag and sound emission were *λ*/*Dm* = 1.892 and *A*/*Dm* = 0.134, with a reduction of 33.21% and 6.331 dB compared to the straight case, respectively. They associated these results with a mean increase in base pressure distribution and a reduction in average vorticity in the near wake.

Finally, a recent relevant study was conducted by Zhang et al. [22], in which LES was used for the comparison of different models at Reynolds number 3900, namely for wavelength ratios varying from 1.8 to 5.0 and amplitude ratios from 0.05 to 0.2. They characterized flow structures developed in the wake as rib-like vortices and three-dimensional vortex lines caused by waviness along the span. Besides that, they also identified longitudinal vortices, and evidenced their similarities with some vortex generator mechanisms. Based on the analysis of two components of the transport equation for vorticity, namely stretching and turning terms, they inferred two vortex formations, which they called primary and secondary vortices. The primary seems to induce the secondary one. The first is dominant while the second tends to disappear with increasing wavelengths. A correlation between longer recirculation lengths and drag reduction is observed for all their wavy cylinders. Drag reduction was found for greater amplitudes and/or smaller wavelengths. Analysis of surface streamlines revealed a flow from the peak to valley before the separation line (curve). The pressure at the stagnation position (leading edge) at peaks and valleys is about the same as the straight cylinder, while the middle section presents a reduction in static pressure, especially for the models with largest amplitude.

Despite being a topic already explored in the literature, others wavy cylinders do not have the same inspiration as the present work. In addition to the geometric transformation applied by the undulation being different, the parameters (wavelength and amplitude) also differ from those typically selected in the wavy cylinder literature. In addition to the presentation of the results, an attempt is also made to establish a correlation that explains the emerging phenomena.

As will be seen later, even though the results were different from what was expected based on the wavy airfoils, since it is a recent and developing topic, the understanding of the phenomenon behind this bio-inspired waviness in a simpler and canonical geometry, such as the circular cylinder, may be useful for other authors to be able to better describe the flow over wavy airfoils or others wavy geometries.

Regarding the wavy cylinders already existing in the literature, it was not expected to find the same results, since they are similar geometries, but not the same, even so, apparently, the phenomenology appears to be common to both. Regarding a possible application of this new geometry, which has a preferential or fixed direction, we can mention rotary wings, such as those found in helicopters, and rotate blades in wind turbines.

This paper is organized as follows. In Section 2, the methodology of the study is presented along with the definition of the numerical domain, the description of the meshes employed and boundary conditions adopted. In Section 3, the numerical results are presented and discussed, starting with a validation of the numerical method as per the results obtained for the straight cylinder, followed by a discussion of the main characteristics found for the wavy models and a detailed assessment of three-dimensional flow structures. Finally, Section 4 summarizes the conclusions of the study.
