4.1. Measured Reynolds number vs drag coefficient data in water

Based on the experimental set-up and methodology noted in § 3.1, measurements of $\bar {C}_{D}$ and $\overline {Re}$ were obtained for $30<\overline {Re}<800$, corresponding to 13 different diameter spheres rolling on four separate panels. Figure 6(a) shows all of the data gathered, with the figure legend indicating the marker shapes corresponding to sphere diameters. Figure 6(b) presents the same data on a log–log scale, with the legend showing the marker colours corresponding to panel types. The data indicate a similar trend to those of previous studies (i.e. $1/Re$ trend, as observed by Carty Reference Carty1957; Garde & Sethuraman Reference Garde and Sethuraman1969; Jan & Chen Reference Jan and Chen1997). Results of Carty (Reference Carty1957) are plotted in figure 6 for comparison. There is a notable variation of $\bar {C}_{D}$ with panel type and sphere diameter at a given $\overline {Re}$ for $\overline {Re}<200$; at $\overline {Re}=100$ this observed deviation is of the order of $10\,\%$.

Figure 6. Variation of $\bar {C}_{D}$ with $\overline {Re}$ for a sphere rolling down an inclined plane. Markers corresponding to individual sphere diameters are as indicated in the legend of figure 6(a) and the marker colours correspond to the panel as indicated in figure 6(b). (a) All data points, $\overline {Re} = 30\unicode{x2013}800$ range; (b) $\overline {Re} = 30\unicode{x2013}1000$ range in log–log scale.

A distinctly higher $\bar {C}_{D}$ for the three larger CA spheres ($D=6.89$ mm, $9.74$ mm and $11.49$ mm) is seen in figure 6. Generally, these spheres follow a separate $\bar {C}_{D}$ vs $\overline {Re}$ trend compared with the other spheres investigated. Additionally, in figure 9, we observe a distinct increase in $\bar {C}_{D}$ for the $D=6.89$ mm CA sphere, for all four panels considered. At this stage, we cannot quantitatively explain this jump in $\bar {C}_{D}$ for the larger CA spheres. However, the surface finish of these larger spheres (matte finish) was distinct from the smaller spheres (shiny finish). The matte finish spheres contained a higher density of larger peaks and valleys. Despite the distinct surface finishes, the roughness statistics of both types of spheres are nominally similar (table 4, $R_q \approx 0.5\, \mathrm {\mu } {\rm m}$). A plausible explanation, to be investigated in more detail in the future, is that the higher $\bar {C}_{D}$ is due to an increased rolling resistance induced by the higher density of the taller peaks. Mechanisms concerning rolling resistance are discussed further in § 4.4.3.

The results of foam spheres are not included in figure 6, and are presented in § 4.2.

Upon closer examination of the data for $\overline {Re}<300$, multiple trend lines are observed as the panel surface roughness and sphere diameter are varied, as was observed by Houdroge et al. (Reference Houdroge, Zhao, Terrington, Leweke, Hourigan and Thompson2023). This variation of $\bar {C}_{D}$ with panel type is clearly observed in figure 6(b). Figure 7 presents the variation of $\bar {C}_{D}$ with $\overline {Re}$ ranging from $50<\overline {Re}<120$, in which two least-squares lines of the form $a+b/Re$ have been fitted through the data points corresponding to the two panels used. The $r^{2}$ values, indicating goodness of fit of the curves, are approximately $0.9$, indicating a good fit. The deviation of $\bar {C}_{D}$ with the change in panel roughness is highlighted in this figure. There is an approximately $10\,\%$ increase in $\bar {C}_{D}$ at $\overline {Re}=100$ between the frosted glass panel and the glass panel (which corresponds to an 80 times decrease in panel $R_{a}$ roughness). The increase in $\bar {C}_{D}$ with a decrease in roughness (or $\xi =G/D$) can be attributed to the increase in the gap drag component with a decrease in gap height, as was predicted in (2.4).

Figure 7. Variation of $\bar {C}_{D}$ with $\overline {Re}$ for the two individual panels, with least-squares lines fitted through each panel. Marker shapes correspond to sphere diameter, as indicated in the legend of figure 6(a). The $R_{q}$ roughness of the two panels are as follows: frosted glass panel; $2.33\,\mathrm {\mu }$m, glass panel; $0.029\,\mathrm {\mu }$m. Error bars indicate combined bias error and random error. Refer to Appendix A for details on error analysis.

This increase in $\bar {C}_{D}$ with a decrease in panel surface roughness is further highlighted in figure 8, where the sphere diameter was fixed ($D=5.86$ mm) while the panel roughness was varied. The individual variations of $\overline {Re}$ vs $\bar {C}_{D}$ are clearly observed in this figure.

Figure 8. Variation of $\bar {C}_{D}$ with $\overline {Re}$ for a fixed diameter, $D=5.86$ mm. Data for three individual panels, with least-squares lines fitted through each panel are shown. Error bars indicate bias error only.

Upon even further examination of figure 7, what initially appears to be scatter around curves fitted for a panel, more trend lines indicating variation in $\bar {C}_{D}$ vs $\overline {Re}$ for specific diameters are observed. Figure 9 presents the data separated by panel roughness, with least-squares lines of the form $a+b/Re$ fitted through the data corresponding to individual diameters. A subset of the data is presented in this figure to highlight the variation of $\bar {C}_{D}$ with sphere diameter. It is observed that variations in $D$ with a fixed panel surface roughness lead to variations in measured $\bar {C}_{D}$. As indicated in figures 9 and 10, an increase in sphere $D$ leads to an increase in $\bar {C}_{D}$. This observed increase is greater at lower $\overline {Re}$ than at higher values; however, the observed variations still maintain the overall $1/Re$ behaviour. A similar observation was made by Houdroge et al. (Reference Houdroge, Zhao, Terrington, Leweke, Hourigan and Thompson2023) on the variation of $C_{D}$ with the sphere diameter. These observations provide further evidence that the relative roughness ($\xi$) is dependent on the panel roughness, the sphere roughness and also the sphere diameter, as we have assumed in (2.5). For each $\xi$, we observe a separate $\bar {C}_{D}$ vs $\overline {Re}$ relationship.

Figure 9. Variation of $\bar {C}_{D}$ with $\overline {Re}$ for the four panels used for experimentation in the range $100>\overline {Re}>300$. Least-squares lines of the form $a+b/Re$ have been fitted through data that correspond to individual diameters of the spheres used. Error bars indicate bias error only. (a) Glass panel, (b) acrylic panel, (c) ceramic panel, (d) frosted glass panel.

Figure 10. Comparison of $\bar {C}_{D}$ vs $\overline {Re}$ between acrylic and CA spheres, of similar diameters, rolling on a glass panel against analytical prediction (2.4). (a) Two acrylic spheres, (b) two CA spheres.

Figure 10 presents a comparison between the experimentally measured drag coefficients, and the drag coefficients predicted using (2.4) under the assumption $G/D = \xi _{q}$. Figure 10(a) shows that the measured $\bar {C}_{D}$ is in good agreement with analytical predictions for acrylic spheres, and therefore $\xi _{q}$ is a suitable estimate of the effective gap. However, for CA spheres (figure 10b), analytical predictions assuming $G/D = \xi _{q}$ underestimate $C_{D}$, particularly at lower $\overline {Re}$ ($\overline {Re}<100$). Nevertheless, at higher $\overline {Re}$, a better agreement between analytical predictions and experimental measurements is observed. This observation is further supported through data presented in §§ 4.3 and 4.4, where the effective values of $\xi$ required to match the predicted drag (2.4) to the experimental measurements are closer to the measured $\xi _{q}$ for acrylic spheres but are below the measured $\xi _{q}$ for CA spheres. Therefore, the assumption $G/D = \xi _{q}$ is applicable to acrylic spheres, but not CA spheres. As discussed in § 2.2, simple roughness statistics such as $R_{q}$ may not capture the complexities of the various scales of surface roughness that contribute to the effective gap. We also note that the effective $\xi$ required to match the predictions and measurements is still of the same order of magnitude as the measured $\xi _q$ for CA spheres. These issues are discussed further in §§ 4.3 and 4.4. For now, we assume $G/D = \xi _{q}$ is a valid approximation, at least to within one order of magnitude.

As discussed previously in § 1, Goldman et al. (Reference Goldman, Cox and Brenner1967) dismissed surface roughness as a possible explanation for the disparity between the analytical prediction and experimental data. However, based on the data observed in figures 6–10, there is a clear correlation between surface roughness and the measured $\bar {C}_{D}$, and therefore surface roughness is at least partially responsible for allowing the sphere to move. Moreover, the $G/D$ required to match the experimental measurements and the predicted drag using (2.4) is of the same order of magnitude as the r.m.s. surface roughness ($\xi _q$), supporting our hypothesis that surface roughness introduces an effective hydrodynamic gap between the sphere and the wall.

4.2. Measured Reynolds number vs drag coefficient data for foam spheres in air

Goldman et al. (Reference Goldman, Cox and Brenner1967) tentatively suggest that cavitation, rather than surface roughness, is responsible for the motion of the sphere. This suggestion is supported by experimental observation of cavitation bubbles appearing close to the point of contact between the body and the plane for heavy bodies in highly viscous fluids (Prokunin Reference Prokunin2003; Ashmore et al. Reference Ashmore, Del Pino and Mullin2005; Seddon & Mullin Reference Seddon and Mullin2006). Cavitation has been observed primarily in the Stokes regime, where the viscous forces are greater than the inertial forces. We have concentrated our efforts so far on the motion of bodies at somewhat higher $\overline {Re}$ (${>}30$), and have not observed any cavitation bubbles during our experiments in water for acrylic and CA spheres ($\beta = 1.2, 1.3$).

To further evaluate the role of cavitation in the drag experienced by rolling spheres, experimental measurements were obtained for foam spheres rolling down an inclined surface in air. If cavitation, rather than surface roughness, were responsible for determining the effective gap, we would expect large differences in the measured drag coefficients between air and water, given that cavitation does not occur in air.

Figure 11 depicts the obtained $\bar {C}_{D}$ vs $\overline {Re}$ variation for foam spheres rolling down an inclined plane in air on acrylic and frosted glass panels. The results of acrylic spheres of similar diameters rolling on the same panels in water are also plotted for comparison. Experiments performed in air, where cavitation cannot occur, display the same trends and values of $\bar {C}_{D}$ as those performed in water. The agreement between the results of measurements conducted in air and in water suggests that cavitation does not affect the motion of the sphere, at least for the experimental parameters considered in this study.

Figure 11. Variation of $\bar {C}_{D}$ with $\overline {Re}$ for a foam sphere rolling down in inclined surface in air. (a) Foam spheres on acrylic panel, (b) foam spheres on frosted glass panel.

The surface roughness of foam spheres is larger than that of the acrylic and CA spheres, as seen in table 4. However, figure 12 shows that the measured $\bar {C}_{D}$ values are similar to those of the other spheres, and follows the general $\bar {C}_{D}$ vs $\xi _q$ trend.

Figure 12. Variation of $\bar {C}_{D}$ with $\xi _{q}$ for three fixed $\overline {Re}$. Values of $\bar {C}_{D}$ at specific $\overline {Re}$ values were calculated using linear interpolation of nearest neighbours. The $C_{D}$ predictions from (2.4) are also plotted for comparison with experimental results. Results of foam spheres rolling in air are also plotted in the same figure.

Terrington et al. (Reference Terrington, Thompson and Hourigan2022) present a general model for predicting the motion of the sphere in both cavitating and non-cavitating flows. They consider three dynamic regimes: contacting motion without slip; contacting motion with slip; and non-contacting motion. Contacting motion with slip occurs at low inclination angles, with the sphere transitioning first to the slip regime, and then to the non-contacting regime, as the inclination is increased. The present study considers angles of inclination below $20^{\circ }$, both in water and air, and no slipping was observed. Therefore, the results of the present study all fall in the ‘contacting motion without slip’ regime. For a different set of experimental parameters, cavitation may be responsible for determining the effective gap.

4.3. Variation of drag coefficient with surface roughness

This subsection investigates the effects of the r.m.s. surface roughness ($\xi _{q}$) on the mean drag coefficient ($\bar {C}_{D}$). Figure 12 plots the variation of the measured $\bar {C}_{D}$ vs $\xi _{q}$ for three constant values of $\overline {Re}$ ($70,100,150$). Here, $\xi _{q}$ was calculated using (2.5) based on the r.m.s. roughness measurements presented in tables 3 and 4. The $\bar {C}_{D}$ values corresponding to each $\overline {Re}$ were calculated using linear interpolation from the nearest neighbouring $\overline {Re}$. Some scatter in data is observed at each $\overline {Re}$, introduced by the uncertainty of both $R_q$ (${\approx }5\,\%$) and interpolated $\bar {C}_{D}$ (${\approx }2\,\%$) measurements. The approximation of $G$ using $R_q$ roughness will also introduce an error, which is more prominent for the CA spheres, and at lower $\overline {Re}$, as discussed in § 4.4.

A line of best fit of the form $a + b\log (\xi )$ has been fitted through the experimental data to indicate the general trend of decreasing $\bar {C}_{D}$ with increasing $\xi _q$. Light shading around curves indicates the uncertainty of the fitted curve, quantified by constructing a $95\,\%$ confidence interval about the regression line, assuming that errors are normally and randomly distributed. The $r^{2}$ values of the curve fits were $\approx 0.4$, indicating that, while variations in $R_q$ roughness account for some of the variation in $\bar {C}_{D}$, it is not accurate for all cases (see § 4.4.2 which discusses these issues further). The results for foam spheres in air are also plotted in the figure. Note that foam data (which have larger uncertainties) have been excluded from the curve fits.

At $\overline {Re} = 70$, there is approximately a $9\,\%$ reduction in $\bar {C}_{D}$ as $\xi _{q}$ is increased from $4\times 10^{-5}$ to $1\times 10^{-3}$ ($25$-fold increase). At $\overline {Re} = 100$ and $150$, the decrease in $\bar {C}_{D}$ for the same change in $\xi _{q}$ is approximately $10\,\%$. The absolute reductions of $\bar {C}_{D}$ at the same $\overline {Re}$ are approximately $0.42,0.36$ and $0.28$, respectively. The experimental uncertainty of measurements of $\bar {C}_{D}$ is typically $\approx 2\,\%$ (see Appendix A). The measured reduction in $\bar {C}_{D}$ for a $25$-fold increase of roughness is well above the experimental uncertainty of the $\bar {C}_{D}$ measurements. This observation, coupled with the relative agreement between the effective $\xi$ and roughness statistics discussed in § 4.4, support the present hypothesis that surface roughness provides the gap required by lubrication theory, and is dependent on both the sphere and panel roughnesses.

The increase in $\bar {C}_{D}$ with decreasing $\xi _{q}$ is consistent with our hypothesis that the effective gap is governed by the height of surface asperities. Figure 12 includes predicted drag coefficients using (2.4) (predictions from Houdroge et al. Reference Houdroge, Zhao, Terrington, Leweke, Hourigan and Thompson2023), under the assumption $G/D = \xi _{q}$ at $\overline {Re} = 70, 100$ and $150$. The computational and analytical solutions correctly predict that $\bar {C}_{D}$ increases as $\xi _{q}$ is decreased; however, they marginally underestimate $\bar {C}_{D}$ compared with the present experimental results. The discrepancy between the predicted and measured $\bar {C}_{D}$ indicates either that the effective gap is smaller than $\xi _{q}$, or that there is some additional source of drag that has not been accounted for in our analysis.

A possible source of this additional drag could be the rolling resistance due to surface roughness. This is discussed further in § 4.4.3.

The dependence of $\bar {C}_{D}$ on $\xi$ is limited to low $\overline {Re}$ values ($\overline {Re}<500$). In figure 12 we observe that the $\overline {Re}=150$ curve is flatter compared with the smaller $\overline {Re}$. The value of $\bar {C}_{D}$ decreases marginally when $\overline {Re}$ is increased further, although the differences in $\xi$ remain unchanged. This observation can be explained using (2.4). The parameter $\xi _{q}$ affects only the gap-dependent drag ($C_{D,{gap}}$), and not the wake drag ($C_{D,{wake}}$). The gap-dependent drag is proportional to $1/Re$ (2.2), and therefore becomes negligible at high $\overline {Re}$. The wake drag coefficient remains $O(1)$ as $\overline {Re}$ is increased (Houdroge et al. Reference Houdroge, Zhao, Terrington, Leweke, Hourigan and Thompson2023), and therefore dominates the total drag at high $\overline {Re}$. Since the wake drag does not depend on $\xi _{q}$, the sensitivity of $\bar {C}_{D}$ to $\xi _q$ decreases as $\overline {Re}$ is increased.

Referring to the Carty (Reference Carty1957) data in figure 1, $\bar {C}_{D}$ remains relatively constant at $\bar {C}_{D}\approx 1$ in the range $10^{3}<\overline {Re}<10^{4}$. At these high $\overline {Re}$, the total $\bar {C}_{D}$ is almost certainly dominated by wake drag, and therefore $C_{D,{wake}}$ is approximately constant for high $\overline {Re}$. We also note a similar trend in behaviour for a sphere in free fall (Rouse Reference Rouse1946), which does not contain either gap or contact forces. It should be noted that, unlike the case of a free-falling sphere, the presence of the panel impacts the wake structures generated, and therefore the converged $\bar {C}_{D}$. That is, for a free-falling sphere, $\bar {C}_{D} \approx 0.4 (10^{3}<\overline {Re}<10^{4})$ (Rouse Reference Rouse1946), while for a freely rolling sphere, $\bar {C}_{D}\approx 1 (10^{3}<\overline {Re}<10^{4})$; the difference is attributed to the effects of the plane on the near wake and wake structures generated by the sphere.

The above observations support our hypothesis that the surface roughness of both the sphere and the panel results in an effective gap between the sphere and the wall when the two surfaces are in contact. The mean drag coefficient can be estimated by assuming the gap is equal to the combined r.m.s. roughness of both the sphere and the wall ($\xi _{q}$).

4.4. Roughness analysis

So far, we have been assuming the effective gap $\xi$ is equal to the combined r.m.s. roughness of the sphere and panel ($\xi _{q}$). As discussed previously (§ 2.2 and briefly in § 4.1), simple statistical measures such as $R_q$ may not capture the various scales of surface roughness. To investigate these effects further, we consider four cases in detail – two acrylic spheres with $D=3.95$ mm and $D=4.71$ mm and two CA spheres with $D=3.94$ mm and $D=4.87$ mm – rolling on a glass surface. Since the roughness of the glass panel is much smaller than that of the sphere by a factor of 10, we expect the gap will be dominated by the roughness of the sphere.

4.4.1. Relationship between effective $\xi$ and $R_{q}$

We define the effective $\xi (=G_{eff}/D)$ as the $G/D$ required to match the predictions of (2.4) with the experimental measurements. These effective $\xi$ are shown in figure 13 for the four spheres, alongside the measured r.m.s. roughness ($R_{q}$), mean roughness ($R_{a}$) and peak roughness ($R_{p}$) of the sphere, normalised by the sphere diameter. Light coloured shading indicates one standard deviation error of the roughness measurements. As observed in the figure, $R_p/D$ values are generally larger and overestimate the gap height. Though $R_a/D$ and $R_q/D$ are similar in magnitude, $R_a/D$ underestimates the gap for the two acrylic spheres. As such, $R_q$ roughness was chosen as the roughness metric that best correlates with the effective $\xi$.

Figure 13. Effective $\xi$ vs $\overline {Re}$ for four spheres rolling on a smooth glass panel. The measured roughness parameters of the spheres are plotted in the figure. Error bands shown by light shading indicate one standard deviation error of roughness parameters. Panels show (a) $D=3.95\,{\rm mm}$ acrylic sphere, (b) $D=3.94\,{\rm mm}$ CA sphere, (c) $D=4.71\,{\rm mm}$ acrylic sphere, (d) $D=4.87\,{\rm mm}$ CA sphere.

In all four cases, we observe that the effective $\xi$ increases with increasing $\overline {Re}$, before reaching an approximately constant value. This approximately constant value is also shown in the figure as $\xi _{mean}$. The fixed gap height at higher $\overline {Re}$ is of the same order of magnitude as the sphere $R_{q}/D$ for acrylic and below $R_{q}/D$ for CA spheres. This figure indicates that $R_{q}/D$ is an excellent approximation for $\xi$ for the acrylic sphere above $\overline {Re}=50$ for the $D=3.95$ mm sphere and above $\overline {Re}=80$ for the $D=4.71$ mm sphere. However, the $R_{q}/D$ for the CA spheres overestimates the gap by a factor of $2$–$3$ for both CA spheres. The parameter $R_{a}/D$ provides marginally better agreement for the two CA spheres, but the difference is too small to conclude that $R_{a}/D$ is a better metric. However, it is promising that the effective $\xi$ is of the same order of magnitude as $R_q/D$ (or $R_{a}/D$) roughness, for all four cases considered here, highlighting that roughness likely provides a satisfactory explanation for the gap required by lubrication theory, while also allowing solid-to-solid contact.

Figure 13 also indicates that the effective $\xi$ is smaller at low $\overline {Re}$, and increases with $\overline {Re}$. The increase in the effective $\xi$ with $\overline {Re}$ is in general agreement with the predictions of Galvin et al. (Reference Galvin, Zhao and Davis2001) and Zhao et al. (Reference Zhao, Galvin and Davis2002). They considered two scales of roughness – a dense covering of small asperities, and a sparse covering of large asperities. Their model predicts an effective gap close to the height of the small asperities at low speeds, but which increases towards the height of the large asperities at high speeds. It should also be noted that the present combined analytical and numerical solution ignores the effects of rolling resistance. Given that rolling resistance due to collisions between surface asperities (see § 4.4.3) will be a function of the plane normal component of the buoyant weight of the sphere ($W_{B}g\cos (\theta )$) (see figure 2), we can expect this to be more prominent at lower angles of inclination ($\theta$), hence lower $\overline {Re}$. This might provide a reasonable explanation for the observed divergence of effective $\xi$ at low $\overline {Re}$; however, this analysis is beyond the scope of the present study.

4.4.2. Comparison of surface textures of the two materials

From figure 13, we see that, for acrylic and CA spheres of approximately equal diameter, the acrylic sphere has a larger effective $\xi$, while the CA sphere has a marginally larger r.m.s. roughness ($R_{q,{sphere}}/D$). Therefore, the r.m.s. roughness alone is not sufficient to fully characterise the effective gap. To explain this difference, figure 14(a,b) compares the surface textures of the two materials corresponding to $D=3.95$ mm acrylic and $D=3.94$ mm CA spheres. Two-dimensional excerpts of the surface roughness measurements are shown in figure 14(c,d). Sections A–A were selected to represent sections with $R_{q}$ similar to that of the mean $R_{q}$ of the spheres, as indicated in table 4 and figure 13. The geometry of the sphere is also indicated for visualisation.

Figure 14. Detailed review of $D=3.95$ mm acrylic and $D=3.94$ mm CA spheres. (a) Three-dimensional surface of $3.95$ mm acrylic sphere. Measurement $R_{q}=0.40$. (b) Three-dimensional surface of $3.94$ mm CA sphere. Measurement $R_{q}=0.54$. (c) Section A–A of (a). Two-dimensional surface profile of $D=3.95$ mm acrylic sphere. (d) Section A–A of (b). Two-dimensional surface profile of $D=3.94$ mm CA sphere.

The surface textures of both acrylic and CA spheres are remarkably different. The acrylic sphere contains two clearly distinct scales of asperities: small asperities which are typically in the range $0.1\unicode{x2013}0.5\,\mathrm {\mu }$m in height that densely cover the sphere, and a sparse distribution of large asperities in the range $2$–$10$ $\mathrm {\mu }$m in height (10 $\mathrm {\mu }$m asperities are not shown in the figure). The CA sphere contains a single scale of asperities that are typically between $0.5$ and $1\,\mathrm {\mu }$m in height, but which can be up to 5 $\mathrm {\mu }$m in height. The CA sphere also contains large valleys, which are of a similar depth to the height of the asperities, while the acrylic does not contain large valleys.

The $R_{q}$ roughness of the $D=3.95$ mm acrylic spheres was measured to be $0.40$ $\mathrm {\mu }$m while for the CA sphere it was $0.60\,\mathrm {\mu }$m, and these are also indicated in figure 14(c,d) for comparison. The measured $R_{q}$ roughness of CA is approximately $50\,\%$ larger than that of acrylic. This is due to the CA sphere containing a high density of moderately large ($0.5$–$1\,\mathrm {\mu }$m) asperities, including both bumps and valleys. While the acrylic sphere contains some very tall asperities, these are balanced out by a large proportion of the sphere containing only small asperities, leading to a smaller overall $R_{q}$. The mean effective gap (mean $G$) calculated from figure 13 is also shown in figure 14(c,d). This indicates the effective average gap required to match the predicted $\bar {C}_{D}$ with experimental measurements. While the $R_{q}$ roughness of the CA sphere is greater than that of the acrylic sphere, the effective gap of the CA sphere is below that of the acrylic sphere. For both acrylic and CA, the heights of the surface asperities are not uniform across the surface. There are many regions on the sphere's surface where the size of asperities – and therefore the minimum possible gap – is well below both mean $G$ and $R_{q}$. Similarly, there are many regions where the local asperities are taller than both mean $G$ and $R_{q}$.

Therefore, our assumption that surface roughness produces an effective gap is consistent with the detailed measurements of the surface profiles for both acrylic and CA. The effective gap is larger than the smallest asperities, but smaller than the largest asperities, as is to be expected. Simple statistical measures, such as $R_{q}$, can approximately indicate the effective gap. However, $R_{q}$ does not consider several important factors, such as the spatial distribution of asperities, and the existence of multiple scales of surface roughness. In the case of acrylic, the effective gap and the r.m.s. roughness agrees quite well. However, this does not hold for all materials.

4.5. Wake–structure interactions

As discussed previously in § 1, the value of $C_{D}$ of a freely rolling sphere is dependent on the sphere wake, due to the influence of wake shedding on the down-slope and cross-slope velocities. In addition, wake shedding has been observed to lead to sphere VIV (Houdroge Reference Houdroge2017), which in turn has an indirect impact on $C_{D}$. As such, we have conducted an analysis of the sphere wake–structure interaction, to investigate these effects, and to provide experimental validation of numerically observed critical transitions.

4.5.1. Flow visualisations

A UV-induced fluorescent dye visualisation technique was used to visualise the wake structures behind the freely rolling spheres. High-resolution (Nikon D7100 and GoPro Hero 10) cameras were used to capture images of wake formations and will be compared against previous experimental and computational studies.

The sphere was placed atop the panel using a place holder, ensuring that no dust or small air bubbles were present. A concentrated solution of the fluorescence dye was introduced on to the sphere at rest, using a syringe with a long needle. The whole sphere was coated with the dye. After allowing the water surface to settle due to perturbations caused by the insertion of dye, the sphere was gently released from its rest position. The sphere was allowed to roll a minimum of $20D$ before entering the frame of the video camera, to ensure a steady state was achieved.

We note that these visualisations highlight the integral effect of the wake characteristics of the sphere, especially at higher $\overline {Re}$. Smaller vortices often dissipate quickly and are not captured by the camera, and generally only large vortices are highlighted by this method. These visualisations serve as a valuable tool for understanding the overall behaviour of the wake, emphasising the mean behaviour of vortices. Smaller vortices or fluctuations that contribute to the overall behaviour are not effectively captured by this method. As such, these images are mainly useful in understanding the general behaviour of the wake.

Figure 15 displays the evolution of the wake behind a freely rolling sphere, as $\overline {Re}$ is increased. As discussed in § 1, previous studies have identified two critical $\overline {Re}$ where transitions occur, namely $Re_{c,1}=139$ and $Re_{c,2}=192$. Figure 15(a,b), corresponding to $\overline {Re}=33$ and $\overline {Re}=108$, depict the wake prior to the first transition; the wake is steady and attached to the sphere. Some lateral motion across the slope was observed, even at these low $\overline {Re}$. At $\overline {Re}=108$, we observe the development of the recirculation zone downstream of the sphere and the beginning of the formation of hairpin-like structures in the far wake. However, as $\overline {Re}$ is increased, the recirculation zone expands further downstream, ultimately leading to its detachment. This transition to unsteady periodic flow is first observed at $\overline {Re}\approx 135$ in the present study. This is in excellent agreement with the numerically predicted first critical transition $Re_{c,1}=139$. Figure 15(c,d) indicates the flow states just before and after this observed transition. In figure 15(d), we observed hairpin-like structures being shed periodically into the wake. Figure 15(e) shows the developed state where one hairpin vortex has been shed downstream, another has just been shed into the wake and a third is observed attached to the sphere, still in development. Further to that, it is observed that the vortices are shed from alternating sides of the sphere. As was observed by Leweke et al. (Reference Leweke, Provansal, Ormiéres and Lebescond1999), the tail of the shed vortex is connected to the head of the preceding vortex, indicating the complex coupled interaction between the shed and forming vortices. Also in agreement with Houdroge et al. (Reference Houdroge, Leweke, Hourigan and Thompson2017), the vortices are shed at varying orientations. This alternating and varying orientation of shedding leads to the observed VIV response of the sphere, which is discussed in detail in § 4.6.

Figure 15. Plan view of experimental flow visualisation using UV-induced fluorescent dye technique for $\overline {Re} = 33$–$1607$. Images were captured using Nikon D7100 and Go-Pro cameras and post-processed. The sphere is rolling from right to left. The video recordings are provided as supplementary movie 1 available at https://doi.org/10.1017/jfm.2024.146. Panels show (a) $\overline {Re}=33(\pm 1)$, (b) $\overline {Re} = 108(\pm 1)$, (c) $\overline {Re} = 133(\pm 2)$, (d) $\overline {Re} = 135(\pm 2)$, (e) $\overline {Re} = 166(\pm 2)$, (f) $\overline {Re} = 198(\pm 2)$, (g) $\overline {Re} = 221(\pm 2)$, (h) $\overline {Re} = 238(\pm 3)$, (i) $\overline {Re} = 244(\pm 2)$, (j) $\overline {Re} = 321(\pm 5)$, (k) $\overline {Re} = 733(\pm 7)$, (l) $\overline {Re} = 1607(\pm 15)$.

As $\overline {Re}$ is increased further, these vortices are periodically shed into the wake, as seen in figure 15(e–l). The frequency of shedding increases with $\overline {Re}$. Numerical studies have identified a second transition, $Re_{c,2}=192$, beyond which the wake loses its symmetry. However, in experimental visualisations, even a small perturbation in the initial conditions can break the wake symmetry. It has been difficult to capture a symmetrical wake, due to sphere oscillations and the lateral movement across the plane. We were unable to observe a symmetrical wake even at $\overline {Re}<100$. Perhaps it is also worth commenting that even very weakly perturbed numerical simulations of freely rolling spheres see the wake symmetry broken at $Re\simeq 140$, well below the predicted transition for a fixed rolling sphere (Houdroge et al. Reference Houdroge, Zhao, Terrington, Leweke, Hourigan and Thompson2023).

Figures 16 and 17 depict the temporal evolution of the wake of a freely rolling sphere at $\overline {Re}=166$ and $\overline {Re}=238$, respectively. In figure 16, we observe five hairpin vortices shed into the wake within an 8 s period. From $t=0$ to $t=1$ s, we notice the shedding of one vortex from one side of the sphere to the formation of another vortex on the opposite side. This oscillatory behaviour continues throughout the wake-shedding cycle, and we observe that the size and orientation of these shed vortices also vary. Similarly, in figure 17, we observe the same behaviour, but the shedding is at a higher frequency; five vortices are shed in a 5.5 s period, and the cross-slope deviation of the sphere is more pronounced in this figure. Again, we also notice the variation in orientation and size of the shed vortices; some vortices are larger than others in size. The fluctuating component of pressure generated by these variations in the shedding of vortices drives the observed cross-slope oscillations. Section 4.6 discusses these VIVs in detail.

Figure 16. Plan view of the temporal evolution of the wake behind a freely rolling sphere at $\overline {Re} = 166(\pm 2)$. The sphere is rolling from right to left.

Figure 17. Plan view of the temporal evolution of the wake behind a freely rolling sphere at $\overline {Re} = 238(\pm 3)$. The sphere is rolling from right to left.

The visualisations presented in figures 15–17 are in excellent agreement with the wake structures previously observed by Leweke et al. (Reference Leweke, Provansal, Ormiéres and Lebescond1999). However, at high $\overline {Re}$, as seen in figure 15(l), the laminar hairpin structures are no longer visible in the wake, which is more turbulent and high-frequency shedding occurs. Individual vortices are no longer discernible; however, we observed the emergence of large-scale turbulent structures.

4.6. Vortex-induced vibration

Cross-slope VIV driven by fluctuating side forces has been observed on freely rolling spheres in previous numerical and experimental studies (Houdroge Reference Houdroge2017; Houdroge et al. Reference Houdroge, Zhao, Terrington, Leweke, Hourigan and Thompson2023). The present study experimentally characterises sphere VIV using high-resolution videos of spheres rolling down an inclined plane, which were post-processed using path tracking software (Tracker by Open Source Physics). This enables the extraction of the instantaneous body displacement and velocity, which can be compared with previous predictions and measurements. Similar to the procedure outlined in § 3.1, the spheres were allowed to roll a minimum of $20D$ prior to starting measurements, to ensure the asymptotic state was reached prior to recording data.

Figure 18 shows sphere tracking results for $\overline {Re}=221$, characterised by non-dimensional variables $y/D$ and $x/D$, where $y$ is the cross-slope distance, $x$ is the down-slope distance and $D$ is the sphere diameter. In this case, 10 individual runs were recorded while maintaining the angle of inclination $(\theta )$ and constant diameter $(D)$. The total down-slope ($x$) distance over which the sphere was tracked was $357$ mm ($x/D=56.26$). Over this down-slope distance, the mean absolute deviation ($y=5.51\,{\rm mm} (\pm 0.88), y/D=0.87$) represents a $1.5\,\%$ deviation across the slope. The maximum deviation ($y=22.38\,{\rm mm}, y/D=3.524$) represents a $6.3\,\%$ deviation. Measurements were taken at 30 frames per second, with a total measurement duration of approximately 9.5 s for each run. Light grey coloured markers indicate the unfiltered data with a measurement uncertainty of approximately $1\,\%$. A cubic-spline smoothing technique (Python Scipy (Virtanen et al. Reference Virtanen2020) interpolate.splrep package) was used to filter the raw data. The figure shows that the 10 runs correspond to 10 distinct paths, indicating considerable variability in trajectories, despite nominally uniform initial and boundary conditions. Similar measurements were made covering a range of $\overline {Re}$, with similar trends observed. These experiments were undertaken on a glass panel, which has much lower roughness than the spheres ($O(10)$).

Figure 18. Cross-slope variation for a rolling sphere visualised using non-dimensional variables $y/D$ and $x/D$, where $y$ is the cross-slope distance, $x$ is the down-slope distance and $D$ is the sphere diameter. Path tracking of 10 runs of a sphere of $D=6.35$ mm, at a $\overline {Re} = 221(\pm 2)$ on a glass panel is shown in the figure.

Figure 19 depicts the non-dimensional cross-slope displacement and cross-slope velocity of the 10 runs at $\overline {Re}=221$. The adjusted cross-slope displacement $y'/D$ is shown in figure 19(a), where the linear trend was subtracted from the measured $y/D$ data. This adjustment was made to assist in uncovering any underlying periodicity in the signals. In addition to the displacement, the normalised cross-slope velocity is shown in figure 19(b). Figure 19 also indicates local maxima and minima, the corresponding $St$ for each run and the mean $\overline {St}$. Despite considerable variation, these figures show an underlying periodicity in the displacement and velocity signals. The figures also indicate that the velocity of the sphere is quasi-steady periodic with random fluctuations. These fluctuations result from fluctuations in the cross-slope force coefficients due to vortex shedding. Houdroge (Reference Houdroge2017) calculated the cross-slope force coefficients numerically and found that, with increasing $\overline {Re}$, the side force evolves from being steady to periodic following the first critical transition at $Re\approx 136$. They found that, not too far in excess of that Reynolds number, the force coefficient becomes highly irregular. In fact, Rao et al. (Reference Rao, Passaggia, Bolnot, Thompson, Leweke and Hourigan2012) found that, even for a sphere rolling at a fixed velocity down a slope, the cross-slope force develops a second frequency component at the second transition ($Re\approx 191$), leading to loss of mirror symmetry across the wake. The variation of the sphere path and velocity between the ten runs is evidence of this unsteadiness in the wake. However, the velocity fluctuates around a mean value, which enables the derivation of the mean velocity, which also yields $\overline {Re}$ and $\bar {C}_{D}$. The standard deviation between these individual runs contributes to the random error of the $\bar {C}_{D}$ and $\overline {Re}$ measurements, typically comprising over $50\,\%$ of the total uncertainty.

Figure 19. Cross-slope displacement and velocity corresponding to $\overline {Re} = 221(\pm 2)$, with the displacement adjusted to remove total cross-slope deviation over the sampling period. The figures also indicate the locations of maxima and minima, allowing an estimate of the corresponding $St$ for each run and the mean $St$. The velocity signal has been normalised by the time-mean $U_{y}$ component of the sphere velocity. (a) Adjusted non-dimensional cross-slope displacement ($y'/D$) data. (b) Non-dimensional cross-slope velocity ($U_{y}/\overline {U_{y}}$) tracking data.

That the cross-slope displacement and velocity signals display oscillatory components enables the extraction of the oscillation frequency for each variable. Estimates of the oscillations frequencies of each signal were obtained using a peak detection algorithm, with maxima and minima indicated in figure 19. This also enables the calculation of the sphere Strouhal number $(St=f\kern0.06em D/\bar {U})$, which is discussed in § 4.7. This same analysis was carried out over a range of $\overline {Re}$, and the Strouhal numbers corresponding to cross-slope displacement and cross-slope velocity were derived. The results are discussed in detail in § 4.7.

4.6.1. Effects of surface roughness on VIV

To establish the effects of surface roughness on the VIV response, path tracking data of the same sphere rolling on two panels with widely different surface roughnesses were obtained. Figure 20 compares the $y/D$ vs $x/D$ data for the same sphere ($D= 3.94$ mm, material = CA, $R_{q,{sph}}/D=1.5 \times 10^{-4}$) rolling on a glass panel ($R_{q,{pan}}/D=7.4 \times 10^{-6}$) and a frosted glass panel ($R_{q,{pan}}/D=5.9 \times 10^{-4}$) at fixed $\overline {Re}=100$. The glass panel's roughness is only 5 % of that of the sphere, while the frosted glass panel is a factor of 3.9 rougher. Thus, the frosted glass panel is approximately $80$ times rougher than the glass panel. Even under such a pronounced variation in roughness, the observed sphere cross-slope deviation and oscillations appear similar. The maximum deviations at $x/D=100$ are $\approx \pm 3D$, and $\overline {St}_{y}$ for the two panels are $0.062$ for glass and $0.052$ for frosted glass. In addition, $\overline {St}_{Uy}$ is $0.084$ and $0.077$ for the glass and frosted glass panels, respectively. These measurements indicate that the surface roughness has at best a second-order effect on sphere VIV or sphere trajectories, at least within the range of roughness investigated. This, of course, is consistent with previous studies indicating that the time-dependent outer flow, which induces the time-dependent forcing, is relatively independent of the gap flow (Houdroge et al. Reference Houdroge, Zhao, Terrington, Leweke, Hourigan and Thompson2023). As roughness is increased further, surface asperities may start to influence the sphere path and VIV response. However, such an effect will likely be a function of $\overline {Re}$.

Figure 20. Non-dimensional cross-slope displacement comparison between glass and frosted glass panels at the same $\overline {Re}$. (a) Non-dimensional cross-slope displacement ($y/D$) tracking data on the glass panel ($R_{q,{pan}}/D=7.4 \times 10^{-6}$) at $\overline {Re}=100$. Here, $\overline {St}_{y} = 0.062$ and $\overline {St}_{Uy}=0.084$. (b) Non-dimensional cross-slope displacement ($y/D$) tracking data on the frosted glass panel ($R_{q,{pan}}/D=5.9 \times 10^{-4}$) at $\overline {Re}=100$. Here, $\overline {St}_{y}=0.052$ and $\overline {St}_{Uy}=0.077$.

4.7. Strouhal number calculations

The shedding of hairpin vortices at moderate-to-high $\overline {Re}$ has been numerically observed to lead to fluctuations in the sphere force coefficients in the cross-slope direction (Rao et al. Reference Rao, Passaggia, Bolnot, Thompson, Leweke and Hourigan2012; Houdroge Reference Houdroge2017; Houdroge et al. Reference Houdroge, Zhao, Terrington, Leweke, Hourigan and Thompson2023). These force fluctuations are accompanied by a varying velocity component, which in turn leads to variations of the sphere path from the mean down-slope direction. The relationship between shedding of vortices and fluctuations in the sphere cross-slope velocity and displacement will be investigated in this section. This correlation will be studied by analysing the frequency responses of the sphere kinematics.

In the present study, the Strouhal number $(St=f\kern0.06em D/\bar {U})$ of a freely rolling sphere was derived using two techniques. The first was based on the number of vortices shed into the wake of the sphere, where $f$ is the vortex shedding frequency, $D$ is the sphere diameter and $\bar {U}$ is the mean down-slope velocity. The second employed path tracking of the sphere motion and calculation of the frequency of oscillation using a peak detection algorithm in the cross-slope ($y$) direction.

The first method used the video recordings obtained for wake visualisations shown in figure 15; the number of large-scale wake vortical structures shed per given length of time was determined, leading to an estimate of the wake-shedding frequency $f$ at a given $\overline {Re}$. The number of larger-scale hairpin vortices shed into the wake was manually counted, halved to account for shedding from opposite sides and divided by the measurement time to determine $f$. As defined in § 2, $D$ was measured and $\bar {U}$ was calculated based on the time taken to travel a known distance ($\approx 200\unicode{x2013}400\,{\rm mm}$).

The second method used a peak detection algorithm on the sphere-adjusted cross-slope displacement ($y'/D$) and cross-slope velocity ($U_{y}/\bar {U}_{y}$) to calculate the sphere oscillation frequency ($\,f$). The maxima and minima of the sphere displacement and velocity signals were identified using a peak detection algorithm, added together, halved and divided by the measurement duration to calculate $f$ for each run.

While the displacement and velocity signals should contain the same frequency components, the higher-frequency component dominates the velocity signal, while the lower-frequency component dominates the displacement signal. That enabled the derivation of two parameters. The parameter $St_{y}$ corresponds to oscillations in the sphere cross-slope displacement, while $St_{Uy}$ corresponds to oscillations in the sphere cross-slope velocity. All values presented are mean values obtained by averaging the individual $St$ obtained using the path tracking data discussed in § 4.6.

All three sets of results are indicated in figure 21 and compared against existing numerical predictions and experimental data. Given that the signals have a sizeable chaotic component, these estimates of the underlying periodicity are at best approximate.

Figure 21. Estimates of Strouhal numbers ($St$) of a rolling sphere over a range of $\overline {Re}$. Computational and experimental data from previous studies are plotted for comparison. The present results are obtained using the peak detection method applied to the displacement and velocity signals from the path tracking analysis described in § 4.6.

As can be seen from figure 21, $St$ is a function of $\overline {Re}$. These Strouhal numbers were obtained using flow visualisations within the range $130 < \overline {Re} < 420$. For $\overline {Re}<100$, no shedding was observed. For $\overline {Re} > 500$, the shedding frequency was too high to distinguish individual vortices to enable the calculation of $St$. Inspection of figure 15(k,l) highlights this limitation. The wake is chaotic and unsteady, and the oscillations and lateral movement add further complexity to the wake shedding. As such, $St$ estimates beyond $\overline {Re} = 450$ were not reported.

The $St_{y}$ predictions are approximately constant at $St_y \sim 0.05$, with some scatter observed in the derived dataset. On the other hand, $St_{Uy}$ initially increases approximately linearly with Reynolds number. The numerical predictions of Rao et al. (Reference Rao, Passaggia, Bolnot, Thompson, Leweke and Hourigan2012) for a sphere rolling at constant velocity with a fixed $G/D=0.005$ are also shown in figure 21. In that study, two dominant frequencies were identified, corresponding to the cross-slope and down-slope force signals. The figure also shows these two measured Strouhal numbers from numerical simulations of a freely rolling sphere at a fixed gap (Houdroge et al. Reference Houdroge, Zhao, Terrington, Leweke, Hourigan and Thompson2023). The downstream oscillation occurs first due to shedding of hairpins into the wake, followed by the development of cross-slope oscillation at a higher frequency as the Reynolds number is increased. Interestingly, $St_{y}$ estimates from the present study are in broad agreement with the Strouhal number predicted numerically by Rao et al. (Reference Rao, Passaggia, Bolnot, Thompson, Leweke and Hourigan2012) and Houdroge et al. (Reference Houdroge, Zhao, Terrington, Leweke, Hourigan and Thompson2023) for large-scale shedding of hairpin structures into the wake. The figure also indicates that $St_y$ is independent of surface roughness.

Analysis of flow visualisations combined with path tracking of a sphere just prior to the second critical transition was conducted at $\overline {Re}=166$ and also at a higher Reynolds number ($\overline {Re}=372$), as shown in figure 22. Large-scale hairpin vortices are identified manually, and the manually identified large hairpin vortices are numbered in figure 22. Figure 22 shows a qualitative agreement between wake shedding and sphere VIV. However, it should be noted that other mechanisms may also influence sphere VIV, which are discussed in the latter part of this section.

Figure 22. Combined path tracking and flow visualisations. (a) Path tracking combined with flow visualisation (plan view) at $\overline {Re}=166(\pm 2)$. (b) Path tracking combined with flow visualisation (plan view) at $\overline {Re}=372(\pm 3)$.

At $\overline {Re}=166$, the sphere $St_{y}$ and $St_{Uy}$ are approximately equal and the vortical wake structures as marked by the dye visualisations have been identified and numbered, as seen in figure 22(a). Depending on the orientation of the shedding of the hairpin vortex, the lateral direction of motion of the sphere changes. Typically, hairpin vortices are shed from alternating sides, leading to an average mean path without any deviation in one direction. However, as seen in the figure, the intensity of shedding from both sides is not always equal, which leads to the overall cross-slope deviations as observed in figure 18. At $\overline {Re}=166$, each shedding of a hairpin vortex leads to a change in sphere displacement.

The $St_{Uy}$ values measured in the present study initially increase with $\overline {Re}$, and continue to increase up to $\overline {Re}\approx 400$ before suddenly decreasing to a relatively constant value. There is excellent agreement between $St_{Uy}$ and the wake shedding $St$ at $\overline {Re}<400$. This observation suggests that, below $\overline {Re}<400$, the sphere cross-slope velocity and wake shedding are coupled. The combined path tracking and wake-shedding diagrams at $\overline {Re}=372$ shown in figure 22(b) further support this observation. Here, $18$ hairpin vortices are shed into the wake of the sphere at $\overline {Re}=372$, and we observe peaks in $U_{y}$ that correspond to each hairpin vortex. This observation suggests that the sphere cross-slope velocity is coupled to vortex shedding frequency. Close agreement between the wake shedding $St$ and $St_{Uy}$ further supports this hypothesis. We only observe $7$ prominent peaks in the $y'/D$ data, suggesting that shedding of every hairpin does not lead to a significant change in sphere displacement (or direction), as was observed at $\overline {Re}=166$.

The observed VIV response of the sphere displacement and velocity with wake shedding provides valuable insight into the dynamics of the freely rolling sphere. Figure 21 indicates that the VIV response of a sphere contains two dominant frequencies related to the shedding of vortices. These two frequencies are observed beyond $Re_{c,1}$, the critical $\overline {Re}$ beyond which vortex shedding occurs, giving rise to two branches in the $St$ vs $\overline {Re}$ data. The sphere velocity is more responsive to the higher-frequency signal, continuing to increase with $\overline {Re}$ up to $\overline {Re}\approx 400$. The sphere displacement is affected by the lower-frequency forcing due to large-scale shedding, and is converted into a side oscillation when the body is allowed to move laterally. This frequency remains relatively independent of $\overline {Re}$. Interestingly, beyond $\overline {Re}>400$ the high-frequency signal is no longer observed, and both velocity and displacement are observed to respond to the lower-frequency signal.

To understand the relationship between vortex shedding and cross-slope sphere movement, consider that the cross-slope force is expected to oscillate at the frequency of vortex shedding (Rao et al. Reference Rao, Passaggia, Bolnot, Thompson, Leweke and Hourigan2012). An additional contribution to the cross-slope force from the lubrication region is likely to depend on the cross-slope velocity, and act to dampen cross-slope motion. Finally, cross-slope contact forces are also present. The cross-slope acceleration of the sphere will be the cross-slope force divided by the sphere's mass. The cross-slope velocity and displacement are the time-integrated effect of the cross-slope acceleration. Therefore, the cross-slope velocity and position are likely to oscillate at a frequency determined by the wake shedding. However, the time integration may result in a smoothing effect that removes some local minima and maxima, resulting in a lower frequency in the velocity and displacement profiles than the shedding frequency. From figure 21, we see that the velocity profiles have the same frequency as the wake shedding (for $\overline {Re}<400$). This supports our assumption that fluctuations in the cross-slope motion of the sphere are due to cross-slope forces caused by wake shedding. However, the displacement profile has a lower (and constant) frequency, because integrating the fluctuating velocity removes some of the extrema. In addition, experimental limitations such as image resolution and inaccuracies in tracking and post-processing, are likely to further dampen the higher-frequency displacement signals.

Although the $St$ variation presented in figure 21 provides key insights into the sphere's motion, sources of error contributing to the scatter need to be considered. In addition to the shedding of hairpin vortices, other sources of cross-slope movement also exist, which may influence the sphere's path down the inclined plane. For instance, smaller-scale oscillations in the shear layer marking the re-circulation zone may also contribute to sphere movement. These structures can be observed in flow visualisations shown in figure 15. Local gradients on the panel surface and contact with large asperities on both the sphere and panel could also lead to perturbations to the sphere path. In addition, dust deposited on the sphere or panel surface, and micro-air bubbles although (mostly) systematically removed, may also lead to system perturbations.

Presently, we have observed a qualitative correlation between wake shedding and sphere VIV. However, the exact mechanisms that lead to sphere VIV are presently unknown. Sphere VIV is likely due to a combination of many effects such as sphere inertia, the added mass coefficient and perhaps other unknown factors. It is plausible that some of these mechanisms may dominate the VIV response dependent on the sphere $\overline {Re}$. The detailed exploration of these mechanisms is beyond the present study, and we recommend it as future research.