2.1. Combined analytical–numerical predictions

Assuming the cylinder is rolling without slipping, the total drag coefficient can be determined from the hydrodynamic force and torque coefficients. Assuming quasi-steady motion, the balance of torques applied to the cylinder is

(2.3)\begin{equation} T_y - F_C \frac{D}{2} = 0, \end{equation}

and therefore the effective predicted drag coefficient is written as

(2.4)\begin{equation} {C}_{D, pred, num} = {C}_{F,fluid} + {C}_{T, fluid}, \end{equation}

where ${C}_{F,fluid} = \bar {F}_D/(\frac {1}{2}DL \rho _{f}\bar {U}^{2})$ and ${C}_{T, fluid} = (\bar {T}_y/ {D}/{2})/(\frac {1}{2}DL \rho _{f}\bar {U}^{2})$ represent the hydrodynamic force and moment coefficients, respectively. In this study, (2.4) is used to predict the drag coefficient from numerical simulations.

For two-dimensional flow over a rolling circular cylinder, Terrington et al. (Reference Terrington, Thompson and Hourigan2023) decompose the force and moment coefficients into a gap-dependent term and a gap-independent ‘wake-drag’ term. Here, we express this result as a combined predicted drag coefficient:

(2.5)\begin{equation} C_{D,{pred}(2D)} = C_{D, pred,gap(2D)} + C_{D,pred,wake(2D)} \end{equation}

The gap-dependent drag for an infinite cylinder is obtained using lubrication theory, valid for small gap heights ($G/D \ll 1$) (Merlen & Frankiewicz Reference Merlen and Frankiewicz2011; Terrington et al. Reference Terrington, Thompson and Hourigan2023):

(2.6)\begin{equation} C_{D, pred,gap(2D)} = \frac{8 {\rm \pi}}{Re \sqrt{G/D}}, \end{equation}

whereas the wake-drag term was obtained by Terrington et al. (Reference Terrington, Thompson and Hourigan2023) using numerical simulations for steady-state flow. We obtain the following empirical fit to their numerical data:

(2.7)\begin{equation} C_{D, pred, wake(2D)} = 14.9 - 11.0(Re)^{0.033}. \end{equation}

This equation is valid for $10< Re<200$ (see Terrington et al. Reference Terrington, Thompson and Hourigan2023).

Teng et al. (Reference Teng, Rallabandi, Stone and Ault2022) compute the gap-dependent force and moment contributions for a finite-span circular cylinder, both translating and rotating near a plane wall. When expressed as an effective drag coefficient, their solution is given by

(2.8)\begin{equation} C_{D,pred, gap(3D)} = \frac{8 {\rm \pi}}{Re \sqrt{G/D}} - \frac{4\mathcal{I}}{Re\,AR}, \end{equation}

where the integral $\mathcal {I}$ can be approximated using the following rapidly converging series:

(2.9)\begin{align} \mathcal{I} = \frac{64 }{3 {\rm \pi}} \sinh^{{-}1} \left(\frac{AR}{\sqrt{G/D}}\right) - 16 AR + \sum_{n=1,3,5,\ldots} 64 \exp({-n{\rm \pi}/2AR}) \left(\frac{1}{3n{\rm \pi}} + \frac{2AR}{n^{2} {\rm \pi}^{2}}\right). \end{align}

Equation (2.8) represents the influence of the lubrication flow in a narrow gap between the cylinder and the wall on the total predicted drag coefficient. The first term on the right-hand side is the drag coefficient corresponding to two-dimensional flow (infinite-span cylinder), whereas the second term is a correction representing the effect of a finite aspect ratio. However, this expression does not include the influence of outer-flow effects such as wake formation and shedding. To account for these effects, the total predicted drag coefficient will be given by

(2.10)\begin{equation} C_{D,pred} = C_{D,pred(3D)} = C_{D,pred, gap(3D)} + C_{D,pred, wake(3D)}, \end{equation}

where $C_{D,pred, wake(3D)}$ is a function of both $Re$ and $AR$, but not $G/D$. Unfortunately, analytical or numerical results for $C_{D,pred, wake(3D)}$ have not been reported in the literature.

A detailed numerical study is required to obtain accurate estimates of $C_{D,pred, wake(3D)}$, but is outside the scope of the present work. In the absence of a reliable estimate for $C_{D,pred, wake(3D)}$, we assume it will be approximately the same order of magnitude as the wake drag coefficient for the two-dimensional cylinder flow ($C_{D,pred, wake(2D)}$). This is a rough approximation only and is likely to introduce significant and unquantified errors into our estimate of the wake drag coefficient. Fortunately, at the low and moderate Reynolds numbers considered in the present study, the wake drag represents a relatively small contribution to the total drag (10–20 % depending on $\overline {Re}$), so reasonable predictions of the total drag coefficient may still be obtained.

In addition, we note that the wake drag coefficient for the two-dimensional cylinder flow is $1.5\le C_{D,pred, wake(2D)} \le 3$ (Terrington et al. Reference Terrington, Thompson and Hourigan2023) and the wake drag coefficient for the rolling sphere flow is $1.0\le C_{D,sph, wake} \le 1.5$ (Houdroge et al. Reference Houdroge, Zhao, Terrington, Leweke, Hourigan and Thompson2023) for $10\le \overline {Re} \le 200$. Given that the three-dimensional sphere also exhibits a similar range of wake drag values as the two-dimensional cylinder, our assumption that the three-dimensional wake drag of a cylinder will be similar to the two-dimensional wake drag ($C_{D,pred, wake(3D)} \approx C_{D,pred, wake(2D)}$) is sufficiently accurate. This approximation will enable an effective comparison between measured $\bar {C}_{D}$ and $C_{D,pred}$.

2.2. Relationship between gap and surface roughness

The primary objective of this investigation is to establish the correlation between surface roughness parameters and the effective gap ($G_{eff}$) required to determine the drag force acting on the cylinder. The British Standard Geometric Product Specifications (GPS) - Surface texture: Profile method - Terms, definitions, and surface texture parameters, BS ISO 4287:1997 provides a comprehensive framework to characterise surface roughness. Among these parameters, the most frequently used are the mean absolute deviation, root mean square (r.m.s.) and the peak roughness, which are denoted by $R_{a}$, $R_{q}$ and $R_{p}$, respectively. Figure 2 indicates these parameters for a sample profile from a cylinder. See Gadelmawla et al. (Reference Gadelmawla, Koura, Maksoud, Elewa and Soliman2002) for detailed descriptions and analytical expressions for typical roughness statistics.

Figure 2.

Visual representation of the three roughness statistics on a sample cylinder profile.

Drawing upon these roughness parameters, a new non-dimensional relative roughness denoted by $\xi$ is introduced as follows:

(2.11)\begin{equation} \xi = \frac{R_{panel} + R_{cylinder}}{D}, \end{equation}

where $R_{panel}$ and $R_{cylinder}$ denote the roughness parameters corresponding to the panel and cylinder surfaces, respectively. A subscript is employed to indicate the specific surface-roughness statistic used to compute $\xi$. For instance, $\xi _p$, is the non-dimensional roughness determined using the peak roughness of both the panel ($R_{p,panel}$) and the cylinder ($R_{p,cylinder}$), whereas $\xi _a$ and $\xi _q$ are calculated using the mean absolute deviation and r.m.s. roughnesses, respectively.

Equation (2.11) assumes that the effective gap at the point of contact results from a linear summation of two simple roughness statistics. In general, elementary roughness statistics such as $R_p$ may be insufficient to completely describe the complex nature of the surface, and a linear summation of surface roughness statistics may not fully describe the interaction between two complex surfaces. The accuracy of these assumptions and simplified measurements are discussed in detail in § 4.5.

3.1. Summary of experimental set-up and methodology

The present experiments were conducted in a water tank with a glass panel mounted on an adjustable stainless steel frame. Other test panels with various surface roughnesses were also adopted. The inclination angle ranged from $1.5^{\circ }$ to $23^{\circ }$. Panel flatness was determined by measuring surface height variation at discrete points, revealing that panel non-flatness was negligible compared with the panel's downward slope. The test cylinders were pre-soaked underwater, with air bubbles removed by vibration and stirring, prior to carrying out the experiments. The cylinders were then placed on a collection port on the plane and gently released to minimise water surface disturbances. A waiting period of at least 2 min followed any water perturbation before measurements were taken, ensuring the water had adequately settled. Regular cleaning of the water tank prevented dust or fibre deposition on the panel surface.

The velocity of rolling cylinders was determined by timing their travel over a fixed distance. A minimum of $20 D$ rolling distance was allowed prior to measurements to ensure the cylinders attained their time-mean terminal velocity. Initially, a stopwatch measured the time for a 200 mm distance on the removable panel (constituting $30\,\%$ of data). Later, a system with three laser-based object detectors was introduced for improved accuracy and efficiency ($70\,\%$ of data). The results presented in the present study incorporate both data sets, and an uncertainty analysis in Appendix B accounts for measurement errors from both methods. Specifications of the tested panels and cylinders are provided in tables 1 and 2, respectively, in Appendix A. Table 2 specifies unique identifiers for each group of cylinders used with the same diameter and aspect ratio. These identifiers are used herein to refer to distinct cylinders.

Table 1.

Panel types used as inclined planes are detailed here. Max. deviation is the maximum absolute deviation of surface height measurements from the mean plane. Max. gradient is the maximum cross-slope gradient over a minimum cross-slope measurement distance of 50 mm.

Table 2.

Specifications of cylinders used for experimental evaluation. Each cylinder diameter corresponds to a set of 10 individual cylinders, and 3 measurements of cylinder diameter and 1 measurement of cylinder span was measured for each cylinder. The mean values of diameter and span including the error for each set is shown above. Refer to Appendix B for details on error analysis.

The data outlined in § 4 represent average measurements obtained from eight separate runs using cylinders of similar diameter and density. Furthermore, occasional checks were performed at randomised locations on the curve to confirm the consistency of the data, even when the fluid temperature varied. Table 2 indicates that the uncertainty regarding the cylinder diameter within each group of cylinders was generally below $1\,\%$. To prevent any distortion caused by water absorption, the cylinders and panels were regularly removed from the water tank outside measurement intervals and dried.

The uncertainty in cylinder diameter was used to estimate deviations in roundness. Given that the uncertainties in the diameter cylinders were generally below $1\,\%$, it was assumed that deviations in the roundness of cylinders could be neglected.

Preliminary experiments were conducted with a selection of cylinders at various inclination angles ranging from $2^{\circ }$ to $23^{\circ }$ to examine potential cylinder slippage in our trials. A marker was placed on the surface of the cylinder, and the cylinder rolling motion was then recorded using a digital camera. The calculated rotational speed was compared to the measured linear down-slope velocity, revealing no significant difference between the two velocities (less than $1\,\%$). As such, any slippage between the cylinder and the surface was considered to be negligible.

To establish the effects of cylinder deformation due to the normal contact force, Hertzian contact theory was used. A simple case of a two-dimensional spherical body in contact with a flat plate was assumed, and the formula for plastic deformation is found in textbooks such as Johnson (Reference Johnson1985). For an acrylic cylinder of $D=19.7$ mm rolling at $\theta =2^{\circ }$ on a glass plate, where the largest normal contact force is acting on the cylinder, the calculated deformation of the cylinder was in the order of $10^{-12}$ m. This deformation is small compared with the height of asperities (typically $10^{-6}$ m), hence the effect of cylinder deformation on the gap imposed by surface asperities was deemed negligible.

3.2. Surface roughness measurements

Non-contact surface roughness measurements for both the cylinders and panels were acquired using an optical profilometer, specifically the Bruker Contour GT-I. This instrument is housed at the Melbourne Centre for Nanofabrication (MCN), situated within the Victorian Node of the Australian National Fabrication Facility (ANFF). The measurements were performed with a 50$\times$ magnification, employing the vertical scanning interferometry (VSI) technique. VSI relies on a broadband light source and is particularly accurate for evaluating surfaces that are rough. The obtained measurements are detailed in tables 3 and 4 in Appendix A.

Table 3.

Measured surface roughness values of panels. Values presented are the arithmetic mean of five individual measurements. The measurement area of one measurement is $0.25\times 0.25\,{\rm mm}^{2}$ (12 measurements under $50\times 1$ magnification joined together).

*Measurement area of $10\times 10\,{\rm mm}^{2}$ was used for the rough ceramic panel which was obtained by stitching together 100 profile scans using a stylus profilometer. No waviness correction was used.

Table 4.

Measured surface roughness of cylinders. Values presented are arithmetic mean of four (or more) individual measurements of four separate cylinders of the same diameter. The measurement area of each presented measurement is $0.4\times 0.4\,{\rm mm}^{2}$ for all diameters (24 measurements under $50\times 1$ magnification joined together). All measurements were corrected for cylinder curvature prior to obtaining roughness statistics.

In the present work, roughness statistics, such as $R_p$, were obtained for a minimum of four different samples. Each sample was generated using the combination of 24 individual measurements conducted under $50 \times 1$ magnification of each surface. The reported values represent the average across all samples. This means that the peak roughness ($R_p$) reported in this study is a sample-averaged value, and does not represent the highest peak observed across all samples. In addition to the sample-averaged peak roughness, it is also useful to report the parameter $R_{p, 1\sigma }$, signifying one standard deviation above the sample-mean peak roughness. This parameter will be used to differentiate the gap imposed by two types of roughness textures observed, which will be elaborated on further in § 4.5.

Figure 3 presents the surface roughness measurements obtained from two panels and four cylinders that were employed in the present experimental investigations. The surface roughness profiles of the acrylic panel depicted in figure 3(a) span a range from $-1.0$ to $0.7\,\mathrm {\mu } {\rm m}$, which is indicative of a typically smooth surface. The ceramic panel surface displayed in figure 3(b) displays notably higher asperities and deeper valleys, with some reaching heights as large as $2\,\mathrm {\mu } {\rm m}$ and valleys as deep as $-4.6\,\mathrm {\mu } {\rm m}$, characteristic of a typical rough surface.

Figure 3.

Surface roughness profiles obtained using the optical profilometer, under 50$\times$ magnification: (a) acrylic panel; (b) ceramic panel; (c) D4.7-A2.2 cylinder; (d) D7.7-A1.3 cylinder; (e) D11.7-A0.9 cylinder; (f) D19.7-A0.5 cylinder. The arrow at the top right corner of each panel image indicates the approximate down-slope direction, whereas for cylinder images, the arrow shows the direction of rolling of each cylinder, with respect to the directional surface textures.

The surface roughnesses observed in all four cylinders shown in figure 3 display pronounced directional characteristics, with a series of ridges and valleys aligned parallel to the direction of cylinder rolling. These ridges arise from the fabrication process, where the acrylic cylinders were produced using a lathe. Comparatively, the two smaller diameter cylinders shown in figures 3(c) and 3(d) possess a smoother surface compared to the larger cylinders in figures 3(e) and 3(f). In the case of the larger cylinders, the peaks of the ridge-like structures typically exceed $4\,\mu {\rm m}$, whereas the smaller cylinders exhibit peaks within the $1\unicode{x2013}2\,\mathrm {\mu } {\rm m}$ range.

5.1. Background of wake–structure interaction of cylinders

An isolated stationary cylinder in free flow transitions from steady two-dimensional to unsteady periodic shedding at $\overline {Re}=46$ (Bénard–von Kármán shedding), and further three-dimensional instabilities occur at $\overline {Re}=190$ and $\overline {Re}=260$ (Taneda Reference Taneda1956; Barkley and Henderson Reference Barkley and Henderson1996; Williamson Reference Williamson1996; Henderson Reference Henderson1997). It is also well known that at $\overline {Re} \approx 1200$, the shear layers separating from the cylinder surface becomes unstable, and the boundary layer becomes turbulent before separation beyond $\overline {Re} \geq 3 \times 10^{5}$. Adding cylinder rotation to this problem gives rise to a range of additional modes of shedding and instabilities, dependent on $\overline {Re}$, of which a thorough review has been presented by Rao et al. (Reference Rao, Radi, Leontini, Thompson, Sheridan and Hourigan2015a). Interestingly, at a rotation rate (the ratio of the cylinder surface speed to the relative free stream speed) of unity, the wake transitions are similar to those of an isolated non-rotating cylinder.

The presence of a plane or a boundary near the cylinder acts to stabilise the flow (Taneda Reference Taneda1965), and delays the onset of vortex shedding to $\overline {Re}=100$ for $G/D \le 0.3$ (Lei, Cheng & Kavanagh Reference Lei, Cheng and Kavanagh1999). At much smaller gap ratios ($G/D\leq 0.1$), the vortices are generated from the interaction between the separating shear layer at the top of the cylinder and the secondary vorticity formed by the wall boundary layer (Stewart et al. Reference Stewart, Thompson, Leweke and Hourigan2010a).

The addition of rotation to this system further complicates the flow dynamics. Stewart et al. (Reference Stewart, Thompson, Leweke and Hourigan2010a) and Rao et al. (Reference Rao, Stewart, Thompson, Leweke and Hourigan2011) found that the direction of rolling, prograde or retrograde, acts to either destabilise the flow or delay instabilities, respectively. Houdroge et al. (Reference Houdroge, Leweke, Hourigan and Thompson2020) investigated numerically the wake of a freely rolling cylinder with a small gap ($G/D=0.005$), and discovered that as $\overline {Re}$ increases, the cylinder wake becomes unsteady and leads to cylinder oscillations. Previous studies (Stewart et al. Reference Stewart, Thompson, Leweke and Hourigan2010a; Rao et al. Reference Rao, Stewart, Thompson, Leweke and Hourigan2011; Houdroge et al. Reference Houdroge, Leweke, Hourigan and Thompson2017) have found that a uniformly rolling cylinder transitions from steady to unsteady vortex shedding at $\overline {Re}=88$. Rao et al. (Reference Rao, Thompson, Leweke and Hourigan2015b) found numerically that the presence of the wall alters the centreline symmetry of the wake, and as such significantly modifies the transitions. When the gap height approaches small values ($G/D \approx 0.2$), they observed that the steady flow transitions to mode E instability prior to the transition to unsteady base flow. At such gap heights, more common instabilities, such as modes A and B, were not observed in the $\overline {Re} < 400$ range. We note that mode A is three-dimensional spanwise instability with an approximate four-diameter wavelength, whereas mode B has a shorter (one-diameter) spanwise wavelength (Williamson Reference Williamson1996).

Pirozzoli et al. (Reference Pirozzoli, Orlandi and Bernardini2012) investigated the flow around a wheel of $AR=0.4$ at various $Re$, and observed steady laminar flow up to $Re\approx 300$, followed by unsteady planar symmetric flow, with the shedding of hairpin vortices up to $Re\approx 400$. The first transition from periodic, planar symmetric flow to quasi-periodic, asymmetrical flow was found to occur at $Re\approx 500$, with further increase in $Re$ leading to increase in turbulence in the wake of the rolling cylinder. Pirozzoli et al. (Reference Pirozzoli, Orlandi and Bernardini2012) also attributed the formation of the hairpin vortices to a combination of spanwise, streamwise and wall-normal vorticity around the rolling cylinder. Through iso-surface vorticity visualisations, they have shown that spanwise vorticity forms the upper part of the hairpin vortices, whereas streamwise vorticity provides the legs and wall-normal vorticity contributing to the necks. Although these critical transitions have been observed numerically for a cylinder of $AR=0.4$, variations in cylinder $AR$ may influence the $\overline {Re}$ at which these transitions occur. We experimentally investigate these phenomena further in the present study.

Wang et al. (Reference Wang, Zhu, Zhou, Bao, Ping, Han and Xu2020) studied numerically (three-dimensional) the wake of an infinite cylinder near a moving wall, at $G/D=0.2$ and $150 < \overline {Re} < 300$. They observed three different modes of shedding and concluded that the transition to chaotic flow at high $\overline {Re}$ occurs due to the strong nonlinear interaction between these modes with different spanwise wavelengths. We expect similar spanwise modes for the large $AR$ cylinders for the freely rolling case studied here; however, cylinder end effects will also contribute to the wake structures formed.

Houdroge et al. (Reference Houdroge, Leweke, Hourigan and Thompson2020) reported on the Strouhal number, $St$, of two-dimensional freely rolling cylinders, and observed $St$ decreasing from $0.065$ at $Re =100$ to $0.051$ at $Re=300$ for $G/D=0.0003125$. With increasing $G/D$, $St$ was observed to increase at a given $Re$. A similar observation was made for the three-dimensional case, with a weak dependence on the mass ratio of the cylinders. Stewart et al. (Reference Stewart, Thompson, Leweke and Hourigan2010a) also observed $St$ for the two-dimensional cylinders within the same range, with $St$ increasing with decreasing rotation rate of the cylinder. The largest $St (\approx 0.1)$ was observed for a non-rotating cylinder at $160< Re<200$. Experimental flow visualisations obtained in this study are compared against these numerically observed values.

5.2. Variations in the cylinder wake with $\overline {Re}$

Figure 14 displays the evolution of the wake behind a freely rolling cylinder with a fixed $AR\approx 1$, as $\overline {Re}$ is increased. As observed in figures 14(a) and 14(b) at lower $\overline {Re}$, the wake of the cylinder is steady and attached. The shear layer separating from the top of the cylinder forms a recirculation zone comprising a counter-rotating vortex pair which is attached to the cylinder. As the cylinder moves down the panel, this recirculation zone oscillates, causing some minor fluctuations in the cylinder path. Figure 14(c) presents the first instance that we observed unsteadiness in the wake of the cylinder. This corresponded to the detachment of the shear layer from the top of the cylinder to form hairpin vortex loops. This image indicates that the first critical transition to unsteady periodic flow occurs in the range $95<\overline {Re}<116$. The observed hairpin loop appears symmetrical. From figures 14(d) to 14(j) as $\overline {Re}$ is increased, the wake of the cylinder becomes progressively more unstable and chaotic. The frequency of shedding of hairpin vortices also increases with $\overline {Re}$, and we observe some degree of cross-slope movement, generated by this shedding.

Figure 14.

Plan view of experimental flow visualisation using UV-induced fluorescent dye technique for a cylinder with $AR\approx 1$ with $\overline {Re}$ ranging from 79–598: (a) $\overline {Re} = 79(\pm 1)$, (b) $\overline {Re} = 95(\pm 1)$, (c) $\overline {Re}=116(\pm 1)$, (d) $\overline {Re} = 134(\pm 2)$, (e) $\overline {Re} = 157(\pm 2)$, (f) $\overline {Re} = 275(\pm 3)$, (g) $\overline {Re} = 317(\pm 4)$, (h) $\overline {Re} = 368(\pm 4)$, (i) $\overline {Re} = 469(\pm 6)$ and (j) $\overline {Re} = 598(\pm 9)$. Images were captured using a GoPro camera and post-processed. The cylinder is rolling from right to left. The video recordings are provided as supplementary movie 1 available at https://doi.org/10.1017/jfm.2024.833.

Figure 15 visualises the near-wake of a freely rolling cylinder D4.7-A10.7, while $\overline {Re}$ was varied. As the $\overline {Re}$ is increased from $28$–$134$, the near wake becomes more chaotic. At $\overline {Re}=28$, the wake is shorter and no hairpin structures are observed. At $\overline {Re}=45$, we note one hairpin-like structure with increased vorticity in the remaining wake as well. Unlike the $AR=1$ case shown in figure 14, for this cylinder with a higher $AR$, the transition to unsteady periodic shedding of hairpin vortices occurs at a much lower $\overline {Re}$ (${\approx }45$). This indicates that the cylinder $AR$ plays an important role in the wake characteristics of a rolling cylinder.

Figure 15.

Plan view of experimental flow visualisation using UV-induced fluorescent dye technique. Comparison of the near wake of a freely rolling cylinder D4.7-A10.7 with $\overline {Re}$ ranging from 28–134, for a constant D (4.69 mm). Cylinder rolling from right to left. The video recordings are provided as supplementary movie 2 available at https://doi.org/10.1017/jfm.2024.833.

In addition to the streamwise hairpin vortices, there is an increase in spanwise vorticity with increasing $\overline {Re}$. Following the transitions to periodic shedding of hairpin vortices first observed at $\overline {Re}=45$, two distinct structures are observed at $\overline {Re}=50$ and $\overline {Re}=89$. As $\overline {Re}$ is increased further, the wake becomes more chaotic and distinct hairpins are no longer visible; rather, multiple vortices are observed. As discussed by Wang et al. (Reference Wang, Zhu, Zhou, Bao, Ping, Han and Xu2020), the nonlinear interaction of different spanwise modes may contribute to the evolution of the wake to the chaotic state observed at $\overline {Re}=134$. The shape of the near wake also changes with increasing $\overline {Re}$. At low $\overline {Re}$ $(28<\overline {Re}<50)$, the near wake appears to be shorter than the span of the cylinder. However, at higher $\overline {Re}$, the near-wake vortices are observed across its full span.

5.3. Cylinder wake variations with $AR$

In this section, we investigate the variation of a cylinder wake with $AR$. The wake of a three-dimensional cylinder (limited span) differs from that of a two-dimensional cylinder (infinitely long) primarily due to the flow around the ends, more commonly categorised as end effects. End effects are an important consideration for limited-span bodies, where a pressure difference between opposing sides drives the fluid around its ends. To visualise these effects, the wake of a cylinder with a fixed $D\approx 4.7\,{\rm mm}$ and with varying $AR (1.01\unicode{x2013}29.84)$ was visualised while rolling down a plane at a fixed angle of inclination.

Figure 16 shows the images of the near wake of the cylinders. First, even at the same angle of inclination, the induced $\overline {Re}$ of the cylinders generally increases with increasing $AR$. The smaller $AR$ cylinder displays a steady attached wake whereas the larger $AR$ cylinders display a high degree of unsteady shedding of hairpin vortices and turbulence in the wake. The wakes of the cylinders are clearly distinct, indicating that the cylinder $AR$ has a significant effect on the cylinder wake shedding.

Figure 16.

Plan view of experimental flow visualisation using UV-induced fluorescent dye technique. Comparison of the near wake of a freely rolling cylinder, for different $AR$ ranging from 1.01 to 29.84, for a constant $D$ (4.7 mm) and $\theta =14^{\circ }$. Cylinder rolling from right to left. White arrows indicate the shedding of a pair of vortices per shedding cycle. The video recordings are provided as supplementary movie 3 available at https://doi.org/10.1017/jfm.2024.833.

Similar to the observations made by Inoue & Sakuragi (Reference Inoue and Sakuragi2008) for an isolated stationary cylinder, we observe a range of patterns within the wake as the $AR$ of the cylinder is varied. One important difference to the observations made by Inoue & Sakuragi (Reference Inoue and Sakuragi2008) is that the wake is one-sided, due to the presence of the plane wall. However, similar to Inoue & Sakuragi (Reference Inoue and Sakuragi2008), for a low $\overline {Re}\, (67\unicode{x2013}113)$, we observe a counter-rotating vortex pair, which transitions to periodic shedding of hairpin vortices (referred to as type IV shedding by Inoue & Sakuragi Reference Inoue and Sakuragi2008) as $AR$ is increased. As $AR$ is further increased to $19.72$ and $29.84$, we observe some remnants of oblique vortex shedding (type I) with a combination of hairpin vortices. Spanwise wavelengths are also visible for the high $AR$ cylinders; at $AR=29.84$, two pairs of hairpins are shed per shedding cycle. This pair of hairpins is shown in figure 16 using two white arrows for the $AR=29.84$ case.

Another key observation that can be made from figure 16 is the streamwise length of the near wake, compared with its span, decreasing with increasing $AR$. For the first three cases ($AR=1.01,2.16,4.29$), we note that the near wake spans the full length of the cylinder. However, at the higher $AR$ ($AR=10.68,19.72,29.84$), the near wake gradually becomes narrower compared with the cylinder span. That is, we see more fluid flow around the ends to the low-pressure region downstream of the cylinder. This phenomenon could reduce the pressure differential between the front and rear of the cylinder, reducing pressure drag. The near wake of the high-$AR$ cylinders is also increasingly turbulent, with multiple hairpin-like structures observed in the wake. For the lower-$AR$ cases, the near wake comprised a pair of attached counter-rotating vortices, with no hairpin structures shed into the wake.

5.4. Cylinder wake variations with $\xi$

To establish the effects of surface roughness on the cylinder wake, visualisations of the same cylinder (D9.7-A1.0) rolling on two panels with different surface roughnesses were obtained. Figure 17 shows the wake of the cylinder rolling on a glass panel ($R_{p} = 0.308\,\mathrm {\mu }$m) and a frosted glass panel ($R_{p} = 5.96\,\mathrm {\mu }$m), which is $20$ times rougher than the glass panel. As observed in the figure, the wake of the cylinders shows a high degree of similarity and the $St$ are also nearly identical. This observation indicates that the panel surface roughness, or the gap imposed by roughness, has little influence on the wake of a rolling cylinder. The same observation was made from the numerical studies of Merlen & Frankiewicz (Reference Merlen and Frankiewicz2011) and Houdroge et al. (Reference Houdroge, Leweke, Hourigan and Thompson2020).

Figure 17.

Plan view of experimental flow visualisation using UV-induced fluorescent dye technique of the cylinder D9.7-A1.0: (a) glass panel ($R_{p}=0.308\,\mathrm {\mu }$m) at $\overline {Re} = 275(\pm 3)$, $St=0.13$; (b) frosted glass panel ($R_{p}=5.96\,\mathrm {\mu }$m) at $\overline {Re} = 277(\pm 3)$, $St=0.13$. Comparison of flow structures between glass panel and frosted glass panel $R_{p}$ roughness of the frosted glass panel is approximately $20$ larger than the glass panel. The cylinder is rolling from right to left.

5.5. Temporal evolution of the wake of a cylinder

The wake of a freely rolling finite-span cylinder is complex, varying with cylinder $\overline {Re}$ and $AR$. At higher $\overline {Re}$, the wake also exhibits a high degree of unsteadiness which increases with $\overline {Re}$, and the hairpin vortices are shed at a higher frequency. To further investigate this unsteadiness, we have captured the temporal evolution of the wake of three cylinders under varied flow conditions. First, in figure 18, we present snapshots taken of a cylinder D4.7-A10.7 at $\overline {Re}=89(\pm 1)$. Then, the flow around a larger cylinder D9.7-A5.16 ($D=9.7\,{\rm mm}$ and $AR=5.16$) at $\overline {Re}=205(\pm 3)$ is presented. Finally, the flow around a cylinder D19.7-A0.5 at a much higher $\overline {Re}=558(\pm 9)$ is shown to highlight the increased unsteadiness in the wake at higher $\overline {Re}$.

Figure 18.

Plan view of experimental flow visualisation using UV-induced fluorescent dye technique. Temporal evolution of the wake of a freely rolling cylinder D4.7-A10.7 at $\overline {Re}=89(\pm 1)$. The images depict the evolution of the wake from $t=0\,{\rm s}$ ($t^{*}=0$) to $t=6\,{\rm s}$ ($t^{*}=23.3$). Three hairpin vortices are shed into the wake within this time period. Cylinder rolling from right to left. The video recordings are provided as supplementary movie 4 available at https://doi.org/10.1017/jfm.2024.833.

Figure 18 depicts the temporal evolution of the near and far wake of a freely rolling cylinder D4.7-A10.7 at $\overline {Re}=89(\pm 1)$. Snapshots were taken at varied time intervals over three fundamental shedding cycles, both dimensional ($t$) and non-dimensional ($t^{*}=t\bar {U}/D$) timescales have been used to describe the flow. At $t=0$ s ($t^{*}=0$), we observe the initial development of the vorticity within the near wake of the cylinder. Two distinct hairpin structures are observed developing, with their corresponding ends attached to the ends of the cylinder and joined in the middle. The vorticity generated by end effects appears to feed each of the hairpins developed. At $t=1\,{\rm s}$ ($t^{*}=3.9$) and $t=2\,{\rm s}$ ($t^{*}=7.8$), the two structures grow in size and move further away from the cylinder surface. At $t=3\,{\rm s}$ ($t^{*}=11.6$), the individual hairpins intertwine and form a single larger hairpin structure, which is detached from the cylinder and shed into the wake at $t=4\,{\rm s}$ ($t^{*}=15.5$). At the same time, we note that another set of hairpin vortices are being developed, grow and join from $t=3\,{\rm s}$ ($t^{*}=11.6$) to $t=4\,{\rm s}$ ($t^{*}=15.5$) and shed into the wake at $t=6\,{\rm s}$ ($t^{*}=23.3$). At $t=6\,{\rm s}$ ($t^{*}=23.3$), three such vortices are observed in the far wake of the rolling cylinder, and another is still attached to the rear end of the cylinder. These shed vortices move away from the panel and dissipate downstream of the cylinder. In addition, we note that in figure 14, the transition to unsteady vortex shedding occurs at $\overline {Re}\approx 100$; however, at this larger $AR$, this transition appears to occur at a lower $\overline {Re}$.

Figure 19 depicts the temporal evolution of the near and far wake of a freely rolling cylinder D9.7-A5.16 at $\overline {Re}=205(\pm 3)$. This case is at a higher $\overline {Re}$ than figure 18, with a larger $D$ and half the $AR$. Two dyes with different shades of green were used on either end of the cylinder, to visualise the interaction between opposite vorticity generated by opposing ends of the cylinder. We observe that the wake is more chaotic than figure 18; however, hairpin vortices are shed periodically. The streamwise distance between consecutive hairpins is also reduced. The intertwining of the two opposite vortices generated by each end of the cylinder is more clearly observed in this figure. At $t=0\,{\rm s}$ ($t^{*}=0$), we observe the two dyes are coating each end of the cylinder, but as time progresses, there is significant mixing induced by the rolling motion of the cylinder. There is some degree of separation between the two dyes, even in the shed vortices, but as the wakes move downstream, there appears to be a remnant of a single colour.

Figure 19.

Plan view of experimental flow visualisation using UV-induced fluorescent dye technique. Temporal evolution of the wake of a freely rolling cylinder D9.7-A5.16 ($D=9.7\,{\rm mm}$ and $AR=5.16$) at $\overline {Re}=205(\pm 3)$. The images depict the evolution of the wake from $t=0\,{\rm s}$ ($t^{*}=0$) to $t=7\,{\rm s}$ ($t^{*}=16.1$). Two dyes of different shades of green have been used to indicate opposite vorticity generated due to the rolling motion. Cylinder rolling from right to left. The video recordings are provided as supplementary movie 4 available at https://doi.org/10.1017/jfm.2024.833.

Figure 20 depicts the temporal evolution of the near and far wake of a cylinder D19.7-A0.5 at $\overline {Re}=558(\pm 9)$. Shedding of hairpin vortices is pseudo-periodic, with varying angles to the streamwise direction. The cylinder wake is chaotic and clear hairpin structures are no longer observed. However, some resemblance of hairpin structures is noted. The frequency of shedding is further increased compared with figures 18 and 19, and the streamwise distance between each vortex is also reduced. As the vortices move away from the cylinder, they grow in size and dissipate downstream. The vortical structures numerically visualised by Pirozzoli et al. (Reference Pirozzoli, Orlandi and Bernardini2012) at $Re=500$ for a cylinder with a similar $AR$ were compared against the structures observed in figure 20. They were in excellent agreement with the structures observed here. They observed numerically that the alternately shed vortices under these conditions have a tendency to bend to alternating sides in the cross-slope direction, which also corresponds to peaks in the frequency spectra of force coefficients. The shedding of these vortices can be used to calculate the $St$ of the cylinders, which is presented in § 5.5.1.

Figure 20.

Plan view of experimental flow visualisation using UV-induced fluorescent dye technique. Temporal evolution of the wake of a freely rolling cylinder D19.7-A0.5 at $\overline {Re}=558(\pm 9)$. The images depict the evolution of the wake from $t=0\,{\rm s}$ ($t^{*}=0$) to $t=8\,{\rm s}$ ($t^{*}=12.1$). Cylinder rolling from right to left. The video recordings are provided as supplementary movie 4 available at https://doi.org/10.1017/jfm.2024.833.

It should be noted that in experimental visualisations, even a small perturbation in the initial conditions can break the wake symmetry. It has been difficult to capture a purely symmetrical wake, due to cylinder oscillations and the lateral movement across the plane. Pirozzoli et al. (Reference Pirozzoli, Orlandi and Bernardini2012) observed a symmetrical wake up to $Re=400$; however, we have not been able to observe that experimentally.

5.5.1. Strouhal number ($St$) calculations

The shedding of hairpin vortices into the wake of a freely rolling cylinder allows the calculation of the Strouhal number $(St=fD/\bar {U})$. Here, $f$ is the vortex shedding frequency, $D$ is the cylinder diameter and $\bar {U}$ is the mean down-slope velocity.

Figure 21 depicts the variation of cylinder $St$ with $\overline {Re}$ for cylinders of varied aspect ratios (0.5–29) and diameters. The numerical results of Houdroge et al. (Reference Houdroge, Leweke, Hourigan and Thompson2020) for a two-dimensional cylinder with $G/D=0.0003125$ and two-dimensional results of Stewart et al. (Reference Stewart, Thompson, Leweke and Hourigan2010a) for $G/D=0.004$ are also shown in the figure. Three-dimensional numerical $St$ determined from the peak $St$ from the lift coefficient time histories given by Pirozzoli et al. (Reference Pirozzoli, Orlandi and Bernardini2012) for a cylinder with $AR=0.4$ are also indicated in the figure for comparison.

Figure 21.

Comparison of $St$ vs $\overline {Re}$ variation for a freely rolling cylinder with varied aspect ratios (${\approx }0.5\unicode{x2013}30$) and diameters. For experimental data, Stewart et al. (Reference Stewart, Thompson, Leweke and Hourigan2010a) and Yang et al. (Reference Yang, Wang, Guo and Zhang2023) the $St$ of vortex shedding in shown. For Houdroge et al. (Reference Houdroge, Leweke, Hourigan and Thompson2020) and Pirozzoli et al. (Reference Pirozzoli, Orlandi and Bernardini2012) the $St$ of oscillations is indicated. Numerical predictions from previous literature have also been plotted for comparison.

The experimental $St$ of the present study shows a clear increasing trend with increasing $\overline {Re}$. There is general agreement between the results of the present study and numerical results in the range $100<\overline {Re}<200$, but diverge beyond $\overline {Re}>200$. Experimental results indicate an increasing $St$ with increasing $\overline {Re}$, whereas the numerical studies predict a decrease in $St$. Three-dimensional effects, which are not effectively captured in the two-dimensional numerical studies by Houdroge et al. (Reference Houdroge, Leweke, Hourigan and Thompson2020) and Stewart et al. (Reference Stewart, Thompson, Leweke and Hourigan2010b), likely play a significant role in this observed divergence. As observed in figure 16, higher $AR$ cylinders display increased unsteadiness in the wake and pronounced wake shedding, even at the same angle of panel inclination and at similar $\overline {Re}$. Experimental data presented in figure 21 contain a large range of $AR$s (${\approx }0.5\unicode{x2013}30$) and cylinder diameters, which also likely contributes to the divergence, and also the scatter observed in the figure. Yang et al. (Reference Yang, Feng and Zhang2022) numerically observed a similar increase in $St$ from $0.12$ to $0.22$ for an increase of $Re$ from $50$ to $500$ for an isolated three-dimensional cylinder for similar $AR$ (also shown in figure 21). We anticipate that the introduction of a plane will alter the shedding behaviour, but will retain the general $\overline {Re}$ dependence. Therefore, the observed increase in $St$ with $\overline {Re}$ in figure 21 is generally within the expected range of $St$ for a three-dimensional cylinder.

Given the complex coupling of contact forces, fluid forces, frictional forces, three-dimensional effects, gap effects and other minor experimental factors such as plate unevenness, dust and air bubbles, the mechanisms that affect cylinder motion and wake shedding are difficult to quantify. Experimentally, we cannot measure the forces acting on a freely rolling cylinder; as such, a detailed explanation of the observed behaviour is not presented here. We simply present the measured $St$ data, which highlight the general behaviour of a freely rolling cylinder. We acknowledge that some of the experimental factors mentioned earlier will affect the presented results. A detailed three-dimensional numerical investigation (including a numerical stability analysis), with sufficiently small gap ratios, of the effects of $AR$ and $\overline {Re}$ on $St$ is required to obtain a deeper understanding of these observed behaviours, and the mechanisms contributing to the observed periodic fluctuations of the freely rolling cylinder problem. To the best of the authors’ knowledge, such an investigation has not yet been reported in the current literature, and it is considered to be beyond the scope of the present study, which we recommend for future work.