1. Introduction
Rotating and stationary bluff body fluid dynamics have interested scientists and engineers for centuries. Early examples of study in this area relate to the aerodynamics of ball sports. Newton (Reference Newton1672) observed that a tennis ball followed a curved path when spin was applied using the racket. Newton theorised that this curvature was due to the difference between the side of the ball where the direction of surface rotation and the free stream flow were aligned and the side where they opposed each other. Magnus (Reference Magnus1852) performed experiments on rotating cylinders and the eponymous Magnus effect was observed. Rayleigh (Reference Rayleigh1877) expanded upon Magnus’ work and sought to explain the curved paths of tennis balls by calculating the Magnus force acting on the body from the pressure distribution (Seifert Reference Seifert2012). Thomson (Reference Thomson1910) stated that the spin of a golf ball gives interest and irregularity to its flight, referencing the earlier works of Magnus and Rayleigh to explain these phenomena.
In addition to the fundamental research interest, the study of flow past rotating cylinders and spheres has several applications. As demonstrated by the historical research efforts, there are applications in ball sports, including baseball, tennis and golf amongst others. Notably, the phenomena of conventional and reverse swing of a cricket ball and the role of the seam are of interest to the aerodynamics community and can be explained using fluid dynamic principles.
Broadly, from Magnus’ experiments in 1852 to present-day research, there has been a trend to study the flow past rotating cylinders rather than rotating spheres. Thom (Reference Thom1926, Reference Thom1934) undertook experiments on rotating cylinders that were the most complete at that time. He studied the effect of Reynolds number, end plates and surface condition, amongst other parameters (Seifert Reference Seifert2012). After Prandtl (Reference Prandtl1904) published his seminal boundary layer theory paper, he too conducted experiments on rotating cylinders, with an emphasis on the Magnus effect and its application to wind-powered ships (Prandtl Reference Prandtl1925). More recently, the fluid dynamics of rotating cylinders has been studied by Mittal & Kumar (Reference Mittal and Kumar2003); computational analysis was used to assess the stability of the flow as a function of non-dimensional rotation rate, defined as the ratio of surface speed to free stream velocity. The focus of aerodynamic research on circular cylinders over spheres may be explained by various factors. Firstly, cylinders are often used as a theoretical springboard or analogue to investigate an aerofoil or wing, for example, when deriving the Kutta–Joukowski theorem (Anderson Reference Anderson2011). Further to this, cylinders are easier to mount in a wind tunnel setting than spheres without interfering with the flow that is being studied. Lastly, flow past a cylinder can be investigated in two dimensions, whereas this is not the case for spheres. For these reasons and others, rotating spheres and their associated fluid dynamics are less frequently studied compared with cylinders; the present research seeks to partially address this inequality.
The flow past a rotating sphere is governed by the Reynolds number,
${\textit{Re}}$
, which gives the ratio of inertial to viscous forces. The Reynolds number
${\textit{Re}}$
is defined as follows, where
$\rho$
is the fluid density,
$U_{\infty }$
is the free stream velocity,
$D$
is the sphere diameter and
$\unicode{x03BC}$
is the fluid dynamic viscosity:
\begin{equation} {\textit{Re}} = \frac {\rho U_{\infty } D}{\mu }. \end{equation}
The non-dimensional drag coefficient,
$C_{\!D}$
, is defined as follows, where
$F_{\!D}$
is the drag force,
$({1}/{2})\rho U_{\infty }^{2}$
is the free stream dynamic pressure and
$A$
is the reference area:
\begin{equation} C_{\!D} = \frac {F_{\!D}}{\frac {1}{2}\rho U_{\infty }^{2}A} .\end{equation}
The pressure distribution over a surface can be normalised using the non-dimensional pressure coefficient
$C_{\!P}$
as follows, where
$P$
is the pressure on the surface and
$P_{\infty }$
is the free stream static pressure:
\begin{equation} C_{\!P} = \frac {P-P_{\infty }}{\frac {1}{2}\rho U_{\infty }^{2}}. \end{equation}
Achenbach (Reference Achenbach1972) conducted an experimental study of flow past stationary spheres where
$C_{\!D}$
was measured and plotted for a variety of
${\textit{Re}}$
. He classified three regimes in the
$C_{\!D}$
versus
${\textit{Re}}$
plot: subcritical, critical and supercritical. Figure 1(a) shows an illustration of Achenbach’s
$C_{\!D}-{\textit{Re}}$
curve for flow past a sphere with the three regimes labelled. Figure 1(a) indicates there is some ambiguity in the demarcation of the onset of the critical flow regime, as it does not necessarily align with the maximum value of
$C_{\!D}$
. The steep reduction in
$C_{\!D}$
during the critical flow regime is known as the drag crisis and was first observed for spheres in 1912 by Gustave Eiffel. Schlichting (Reference Schlichting2017) notes that drag crisis can only be interpreted as a boundary layer effect. In the subcritical regime,
$C_{\!D}$
is approximately independent of
${\textit{Re}}$
(Achenbach Reference Achenbach1972). In this regime, a laminar boundary layer is formed over the front surface of the sphere and separates close to the shoulder (at approximately
$80^\circ$
relative to the stagnation point) as a consequence of the adverse pressure gradient. At a critical
${\textit{Re}}$
, the separated shear layer becomes unstable and rolls into small vortices which trigger a transition to turbulence (Singh & Mittal Reference Singh and Mittal2005; Deshpande et al. Reference Deshpande, Kanti, Desai and Mittal2017). The resulting turbulent shear layer reattaches to the surface downstream of the separation point forming a laminar separation bubble (LSB). The now turbulent boundary layer separates from the surface of the body later than the laminar layer (at approximately
$120^\circ$
relative to the stagnation point) as it is better able to overcome the adverse pressure gradient due to the greater amount of higher momentum fluid (Scobie et al. Reference Scobie, Pickering, Almond and Lock2013). This causes a substantive decrease in pressure drag due to an increase in base pressure and a reduction in wake size (Singh & Mittal Reference Singh and Mittal2005; Schlichting Reference Schlichting2017).
Flow regime classification: (a) variation of drag coefficient with
${\textit{Re}}$
, with flow regimes as per Achenbach (Reference Achenbach1972) and (b) variation of
$\overline {C}_{\!D}{\textit{Re}}^{2}$
with
${\textit{Re}}$
(Schewe Reference Schewe1983; Chopra & Mittal Reference Chopra and Mittal2022).
Schewe (Reference Schewe1983) proposed an alternative method for classifying flow regimes in relation to flow past a circular cylinder. Schewe demonstrated that flow regime classification was unambiguous if the variation of drag force,
$F_{\!D}$
, with
${\textit{Re}}$
was used rather than
$C_{\!D}-{\textit{Re}}$
variation. He suggested that the maximum and minimum
$F_{\!D}$
, respectively, represent the onset and end of the critical flow regime. Chopra & Mittal (Reference Chopra and Mittal2022) drew an equivalence between using drag force and the quantity
$\overline {C}_{\!D}{\textit{Re}}^{2}$
for flow regime classification. They tested the modified method of regime categorisation using Schewe’s time-averaged drag coefficient (
$\overline {C}_{\!D}$
) and
$F_{\!D}$
data and confirmed that the quantity could be used as a proxy for drag force in classifying the flow regime. An illustration of the classification scheme is given in figure 1(b).
The effect of surface roughness on the flow past stationary cylinders was investigated by Achenbach (Reference Achenbach1971), and for stationary spheres by Achenbach (Reference Achenbach1974). In both cases, he experimentally measured local pressure and skin friction distributions to evaluate
$C_{\!D}$
and the angular location of boundary layer separation for different cylinder and sphere roughnesses. Increased surface roughness was found to decrease the critical
${\textit{Re}}$
(earlier onset of drag crisis with respect to
${\textit{Re}}$
). Surface roughness disturbs the laminar boundary layer and causes the transition to turbulence to occur at lower
${\textit{Re}}$
(Scobie et al. Reference Scobie, Pickering, Almond and Lock2013). Limited research has been conducted into the effect of surface roughness on the fluid dynamics of rotating spheres. Barlow & Domanski (Reference Barlow and Domanski2008) tested rotating spheres of differing surface texture and found that this may influence the direction and magnitude of the lift generated by the spin, although they state the impact of surface roughness received only limited examination. The effects of surface roughness on the flow past rotating spheres will be examined further in this study.
A laminar boundary layer separates from the sphere surface due to its inability to overcome the adverse pressure gradient; the point at which this occurs will be referred to as laminar separation (LS). Under certain conditions, the separated shear layer becomes unstable and rolls into small vortices which trigger a transition to turbulence (Singh & Mittal Reference Singh and Mittal2005). The turbulent shear layer has higher momentum than its laminar counterpart and hence is better able to overcome adverse pressure gradients. It thickens in the fluid after separating and eventually reattaches to the surface (Scobie et al. Reference Scobie, Pickering, Almond and Lock2013); this is known as the turbulent attachment (TA) point. Between the points of LS and TA a region of recirculation is formed, known as the LSB (Tani Reference Tani1964). Studies have shown that the signature of the LSB in the
$C_{\!P}$
distribution is a plateau of relatively constant pressure (Achenbach Reference Achenbach1968; Cheng et al. Reference Cheng, Pullin, Samtaney, Zhang and Gao2017; Chopra & Mittal Reference Chopra and Mittal2017). Fage (Reference Fage1936) used surface pressure measurements to identify the LSB; a narrow constant pressure region indicated its presence. Achenbach did not replicate Fage’s findings in 1972 but attributed this to the relatively large step size (
$\Delta \theta =5^\circ$
) for which measurements of surface pressure were taken (Achenbach Reference Achenbach1972). Taneda (Reference Taneda1978) used oil flow visualisation to identify the presence of the LSB; three distinct lines were identified which were correlated to the points of LS, TA and turbulent separation (TS). The LSB is closely linked to the phenomenon of drag crisis (Singh & Mittal Reference Singh and Mittal2005). Deshpande et al. (Reference Deshpande, Kanti, Desai and Mittal2017) performed an experimental study of flow past a stationary sphere with a focus on the intermittency of the LSB. They divided the critical flow regime into three subregimes based on the appearance and activity of the LSB. The LSB was not present in subregime I, and a decrease in
$C_{\!D}$
was found to be due to an increase in mean base pressure (
$\overline {C}_{P,b}$
). Subregime II was characterised by the intermittent appearance of the LSB and a rapid decrease in
$C_{\!D}$
; in subregime III the LSB was found to continually exist. Deshpande et al. (Reference Deshpande, Kanti, Desai and Mittal2017) defined the intermittency factor,
$I_{\!f}$
, as the fraction of time for which the LSB exists in the flow;
$I_{\!f}=0$
in subregime I and
$I_{\!f}=1$
in subregime III. They found that the variation of
$(1-I_{\!f})$
with
${\textit{Re}}$
closely followed the variation of time-averaged drag coefficient (
$\overline {C}_{\!D}$
) with
${\textit{Re}}$
, which demonstrated that the sharp decrease in
$C_{\!D}$
during drag crisis is primarily due to the increased probability of the LSB forming on the surface.
A secondary recirculation region, downstream of the LS point, has been observed in the flow past spheres and cylinders (Son & Hanratty Reference Son and Hanratty1969; Muto, Tsubokura & Oshima Reference Muto, Tsubokura and Oshima2012; Cheng et al. Reference Cheng, Pullin, Samtaney, Zhang and Gao2017; Chopra & Mittal Reference Chopra and Mittal2022; Desai & Mittal Reference Desai and Mittal2022). Son & Hanratty (Reference Son and Hanratty1969) measured velocity gradients on the surface of a stationary cylinder and observed this secondary region of recirculation, which they termed the secondary vortex (SV). Ono & Tamura (Reference Ono and Tamura2008) performed a large-eddy simulation (LES) of the flow past a stationary cylinder and found the LSB and SV to coexist in the flow, with the SV forming between the separation and reattachment points of the LSB. Figure 2(a) shows an illustration adapted from Chopra & Mittal (Reference Chopra and Mittal2022). The SV is highlighted in light pink and the LSB in light blue; SA denotes secondary attachment and SS denotes secondary separation. A comprehensive numerical study of the flow past a stationary circular cylinder by Chopra & Mittal (Reference Chopra and Mittal2022) showed that the SV forms in the subcritical regime and continues to exist in the critical and supercritical regimes. They found that the LSB emerges in the critical flow regime and coexists with the SV in the critical and supercritical regimes, where the SV was embedded within the LSB. Notably, Chopra and Mittal showed there was no discernible signature of the SV in the
$C_{\!P}$
distribution when the flow was devoid of LSB. However, when the SV and LSB coexist, the SV modifies the LSB plateau in the
$C_{\!P}$
distribution to include a sharp dip resembling ‘kink’ followed by a swift recovery – figure 2(b) identifies the SV imprint. Desai & Mittal (Reference Desai and Mittal2022) experimentally investigated the topology of the LSB in the flow past a stationary sphere using surface pressure measurements and oil flow visualisation. They found it impossible to detect SV and LSB coexistence using their experimental techniques as the SV imprint was much weaker than that of the LSB.
LSB and SV: (a) illustration showing the coexistence of LSB and SV in the flow, size has been exaggerated, modified from Chopra & Mittal (Reference Chopra and Mittal2022); (b) indicative
$C_{\!P}$
distribution with the LSB and SV coexisting in the flow, ‘kink’ highlighted by dashed red circle.
The non-dimensional rotation rate,
$\alpha$
, is the ratio of surface speed to free stream speed – see (1.4), where
$\omega$
is the rotational speed in revolutions per second; the present study spans
$0\leqslant \alpha \leqslant 0.45$
. The Magnus effect produces a force on a rotating sphere in a direction normal to the free stream flow and the axis of rotation (Swanson Reference Swanson1961). The Magnus force is primarily governed by
$\alpha$
,
${\textit{Re}}$
and surface roughness. The retreating side of a sphere is where the surface velocity under rotation is in the same direction as that of the free stream flow. When a rotating sphere is experiencing the ordinary Magnus effect, the boundary layer separation location is delayed on the retreating side due to a reduction in local shear relative to the other side (Li et al. Reference Li, Zhang, Liu, Azma and Gao2023a
). The side where the surface velocity opposes that of the free stream flow is known as the advancing side. On the retreating side, where separation occurs farther downstream (at a larger angle from the stagnation point), the pressure is more negative than on the advancing side and a force is generated in the direction towards the retreating side (Swanson Reference Swanson1961). Under certain conditions, a force may be generated in the opposite direction to that caused by the Magnus effect, known as inverse Magnus. Kim et al. (Reference Kim, Choi, Park and Yoo2014) state that this occurs when the flow on the advancing side transitions to turbulence and the retreating side flow remains laminar. The boundary layer separates farther downstream on the advancing side versus the retreating side and the direction of the force is reversed. Sakib & Smith (Reference Sakib and Smith2020) performed particle image velocimetry (PIV) on smooth spheres and golf balls and showed
$C_{L}$
to be proportional to the difference between the separation angle on the retreating side (
$\theta _{\textit{sep}, \textit{ret}}$
) and on the advancing side (
$\theta _{\textit{sep}, \textit{ad}v}$
). They also demonstrated the inverse Magnus effect when
$\theta _{\textit{sep}, \textit{ad}v} \gt \theta _{\textit{sep}, \textit{ret}}$
. The non-dimensional rotation rate is given by:
\begin{equation} \alpha = \frac {\omega \pi D}{U_{\infty }} .\end{equation}
Li et al. (Reference Li, Zhang, Liu and Gao2023b
) studied the flow structure in the wake of a rotating rough sphere with
$\alpha$
ranging from
$0$
to
$6.0$
and
${\textit{Re}}\approx 8000$
. Particle image velocimetry was used in a water tunnel to measure time-averaged flow fields at different downstream planes. A pair of counter-rotating vortices were detected, one on each side of the sphere centre plane. Li et al. (Reference Li, Zhang, Liu and Gao2023b
) found that the flow leakage (downwash) from the advancing to the retreating side, created by the rotation-induced pressure differential, generated these vortices. An equivalence was drawn with the creation of wing-tip vortices downstream of an aircraft. Hoerner (Reference Hoerner1935) and Norman & McKeon (Reference Norman and McKeon2008) investigated the effect of the supporting sting diameter on the flow past stationary spheres. The ratio of sting diameter (
$D_{s}$
) to sphere diameter (
$D$
) is defined as
$\chi =D_{s}/D$
; in the present study
$\chi =0.14$
. The experiments by Li et al. (Reference Li, Zhang, Liu and Gao2023b
) used a set-up with
$\chi =0.15$
and the same sting-to-flow orientation as the present study.
Parekh, Chaplot & Mittal (Reference Parekh, Chaplot and Mittal2024) studied the flow past a cricket ball using LES for
${\textit{Re}}$
$O(10^{5})$
. They found that the pressure differential between the seam and non-seam sides of a non-rotating cricket ball also set up ‘wing-tip-like’ vortices (WTVs). At the downstream axial plane closest to the ball, they noted three pairs of WTV: primary, secondary and tertiary, of decreasing strength, with each pair being divided across the centre plane. Parekh et al. (Reference Parekh, Chaplot and Mittal2024) also observed that the polarity of the WTV reversed when the suction side switched from the seam to the non-seam side at
${\textit{Re}}=4.5\times 10^5$
; these were coined reverse wing-tip-like vortices (RWTVs). Li et al. (Reference Li, Zhang, Liu and Gao2023b
) did not observe a reversal of vortex polarity as tests were conducted for a fixed
${\textit{Re}}$
that was insufficiently large for RWTVs to form. Graftieaux, Michard & Grosjean (Reference Graftieaux, Michard and Grosjean2001) proposed two vortex identification functions,
$\varGamma _{1}$
and
$\varGamma _{2}$
. These dimensionless scalars can be used to identify the centre and extent of a vortex based on the velocity field and will be used in the present study for the analysis of WTV.
This investigation experimentally studies the flow past rotating spheres of varying surface roughness for
$5\times 10^4\leqslant {\textit{Re}}\leqslant 3\times 10^5$
and rotation rate
$0\leqslant \alpha \leqslant 0.45$
. The study addresses the following research questions. (i) Can the coexistent signatures of the SV and LSB in the surface pressure coefficient distribution be experimentally detected in the flow? (ii) What is the effect of surface roughness, rotation rate and Reynolds number on: the appearance of the LSB and SV; and the transition from ordinary to inverse Magnus effect? (iii) Are counter-rotating WTV pairs generated in the wake of a smooth rotating sphere; and does vortex polarity reverse upon the transition from ordinary to inverse Magnus effect?
2. Experimental set-up and methodology
2.1. Experimental rig details and coordinate system
Two spheres with a diameter
$D=143$
mm with differing surface roughness were tested. The spheres were comprised of two additively manufactured hemispherical shells so that rotating pressure transducers could be placed inside. A
$90^\circ$
spline of holes for pressure taps was included in one hemisphere, 21 holes with
$4.5^\circ$
angular spacing. For the ‘rough’ hemispherical shells, no further surface finishing operations were carried out and the surface was left as manufactured. The ‘smooth’ hemispheres were sanded with low-grit sandpaper, coated with polyethylene paint and then sanded with high-grit sandpaper. Stainless steel tubing (
$1.10$
mm outer diameter) was used to create the pressure taps.
The centre of the sphere is considered the origin; results will primarily be reported on the equatorial plane where the two hemispheres join (
$YZ$
plane at
$x=0$
). The azimuthal angle
$\theta$
denotes the angular position along the equator and the polar angle
$\phi$
denotes the angular position along the meridian;
$\theta =\phi =0^\circ$
at the stagnation point. Figure 3 shows these planes in the context of the coordinate system and the stagnation point, identifying the azimuthal and polar angles. The sphere rotates about the
$x$
-axis where
$\omega$
is the rotational speed in revolutions per second. The free stream flow is positive in the
$y$
-direction. Figure 4 illustrates the advancing and retreating sides of the sphere and defines the azimuthal angle,
$\theta$
, on each side.
A view of the sphere defining the coordinate system, azimuthal angle (
$\theta$
), polar angle (
$\phi$
) and locations of stagnation points and pressure taps. The sphere is formed from two hemispheres joined along the equator (dash–dot line) and rotates about the
$x$
-axis.
Schematic illustrating the direction of free stream flow and the advancing and retreating sides of the sphere. Azimuthal angles are defined from
$\theta = 0 ^\circ$
at the stagnation point moving anticlockwise.
(a) Annotated isometric view of the rig, and (b) rig with frame and base plate in the open-jet wind tunnel; nozzle shown in blue, collector removed for clarity.
Indicative processed surface roughness profiles showing
$z$
displacement (
$\unicode{x03BC}$
m) of the stylus tip normal to the surface across the sample length (mm) for (a) smooth and (b) rough spheres.
Throughout the experiments the presence of the shaft, oriented perpendicular to the free stream, means the flow is not symmetric in the
$XZ$
plane. The effect of the shaft on pressure measurements is discussed in more detail in § 3.1 and the effects on WTV formation in the sphere wake are discussed in § 3.3.
Experiments were carried out in the open-jet wind tunnel in the Department of Mechanical Engineering at the University of Bath. The tunnel had a working section of length 1.5 m, a circular nozzle of diameter 0.76 m and a collector of diameter 1.1 m. Free stream turbulence intensity was less than 1 %. The test rig can be seen labelled in isometric view and in the context of the wind tunnel in figures 5(a) and 5(b), respectively. Ten Honeywell TruStability SSC Series differential pressure transducers were housed inside the sphere with a pressure range of
$\pm 10$
mbar and an accuracy of
$\pm 0.2$
mbar. The rotating shaft was protected by a non-rotating sleeve of diameter
$D_{s}=20$
mm. An external ESI PR3202 differential pressure transducer with a pressure range of
$\pm 15$
mbar and an accuracy of
$\pm 0.045$
mbar was connected to a pitot-static tube in the wind tunnel nozzle to record time synchronised free stream dynamic pressure (
$P_{\textit{stagnation}}-P_{\textit{static}}$
). The pressure transducers were calibrated using a Fluke 718 1G pressure calibrator over an applied range of
$\pm 4$
mbar.
2.2. Surface roughness measurement
Surface roughness was measured using a Taylor Hobson Intra Touch profilometer. Arcs across the hemisphere surface were traversed by the profilometer stylus tip. Measurements were taken using a stylus velocity of 0.5 mm s−1. Data were fitted to a least squares arc; a modified profile was obtained which plotted the movement of the stylus tip normal to the surface in micrometres.
Figures 6(a) and 6(b) show indicative examples of the roughness profiles for the smooth and rough hemispheres, respectively. The smooth hemisphere exhibits irregular roughness fluctuations across the sample length (2–3
$\unicode{x03BC}$
m amplitude), this is consistent with a painted surface with no regular roughness pattern. The rough hemisphere shows a regular pattern of larger roughness fluctuations (15–30
$\unicode{x03BC}$
m amplitude), this is consistent with the profile expected from an additively manufactured part with distinct layers.
For each sample, a surface roughness analysis was performed to obtain an arithmetic mean roughness (Ra) value; table 1 presents the results. The average Ra value for the rough sphere (Ra, 10.7
$\unicode{x03BC}$
m) was an order of magnitude larger than that of the smooth sphere (Ra, 1.3
$\unicode{x03BC}$
m).
Values of average arithmetic mean roughness (Ra) for the ‘smooth’ and ’rough’ spheres.
2.3. Surface pressure measurement
Surface pressure data were recorded for approximately 10 s at an acquisition frequency of 15 kHz for all rotating experiments. Table 2 summarises the test parameters for the rotating pressure measurements collected for both spheres, where rotational speed (
$\omega$
) was fixed rather than non-dimensional rotation rate (
$\alpha$
). For each test condition, the measurements were collected for 20 of the 21 pressure taps. Given that there were 10 internal pressure transducers, data were measured in two parts using two different pressure tap/transducer configurations.
Test matrix showing parameters for pressure measurement tests.
Experimental configuration of the cross-flow PIV set-up, adapted from Jackson et al. (Reference Jackson, Harberd, Lock and Scobie2020).
2.4. Cross-flow PIV
Cross-flow PIV images were captured on
$XZ$
planes using an eight megapixel TSI Incorporated charged coupled device camera with a 100 mm focal length lens. Free stream flow was seeded with oil particles using a six-jet droplet generator, producing an average droplet diameter of 1
$\unicode{x03BC}$
m. Tracer particles were illuminated using a 120 mJ double pulse Nd:YAG laser. The camera and laser were controlled using a TSI 610034 synchroniser and the TSI Insight 4G software package. The time step between consecutive laser pulses was set to
$\Delta t=10$
$\unicode{x03BC}$
s and PIV exposure to
$1500$
$\unicode{x03BC}$
s. The camera was mounted in the tunnel outlet as per the experimental set-up shown in figure 7. The majority of image acquisition was phase-locked with sphere rotation to minimise the effects of eccentricity. Due to equipment limitations, PIV images acquired when pressure data were being simultaneously collected could not be phase-locked and consequently, the pulse rate was set to 1 Hz during these tests. Images were captured on two axial
$XZ$
planes downstream of the sphere, defined by the non-dimensional distances,
$y/D=1.5$
and
$2.0$
, where
$y=0$
at the sphere centre. The field of view at
$y/D=1.5$
was 246 mm by 184 mm in the
$x$
- and
$z$
-directions, respectively, and at
$y/D=2.0$
was 233 mm by 174 mm. For each test condition, up to 500 image pairs were captured and processed using a recursive approach based on the Nyquist frequency in TSI Insight 4G. Time-averaged velocity fields were produced for each test. PIV was performed only in the wake of the smooth sphere for the conditions outlined in Table 3.
3. Results, analysis and discussion
3.1. Effects of the shaft and validation of pressure coefficient distributions
In order to first determine the influence of the supporting shaft, experiments were conducted where the angle between the shaft and free stream flow was varied. This was adjusted by bolting the support stand in different positions on the base plate (the holes can be seen in figure 5
b). The position of the shaft was defined using
$\beta$
, the offset between the angle of the shaft and
$x$
-axis. When the shaft was moved towards the free stream flow,
$\beta$
was defined as positive. Here
$\beta =0^\circ$
when the shaft was oriented perpendicular to the free stream flow.
Variation of
$\overline {C}_{\!P}$
distribution with shaft angle (
$\beta$
) on the
$XY$
(meridian) plane, at
${\textit{Re}}=1.16\times 10^{5}$
and
$\alpha =0.19$
.
Test matrix summarising PIV testing.
Figure 8 presents the variation of time-averaged pressure coefficient (
$\overline {C}_{\!P}$
) with
$\beta$
at
${\textit{Re}}=1.16\times 10^{5}$
and
$\alpha =0.19$
, on the
$XY$
(meridian) plane. The shaft angle,
$\beta$
, was varied between
$\beta =30^\circ$
(towards the free stream flow) and
$\beta =-30^\circ$
. Close agreement in
$\overline {C}_{\!P}$
profile is shown across all shaft angles. Whilst the presence of the shaft leads to flow asymmetry in the
$XZ$
plane, figure 8 provides evidence that flow phenomena on the pressure tapped hemisphere (non-shaft side) are insignificantly affected by the shaft (and its position relative to the free stream flow) in the current experimental set-up. Here
$\beta =0^\circ$
was used for all further surface pressure measurements presented in this study.
Given the novel experimental approach of collecting surface pressure data for rotating spheres, validation of the experimental set-up was considered necessary. Figure 9 presents comparisons of
$C_{\!P}$
distributions from the present research compared with those from prior studies, for both stationary and rotating conditions. Figure 9(a) compares the pressure coefficient distributions in the
$XY$
(meridian) plane for the stationary smooth sphere from the present study with previous research by Fage (Reference Fage1936), Achenbach (Reference Achenbach1972) and Muto et al. (Reference Muto, Tsubokura and Oshima2012). Good agreement is shown both quantitatively and qualitatively between the present experimental results at
${\textit{Re}}=1.7\times 10^{5}$
and data from Achenbach (Reference Achenbach1972) at a similar
${\textit{Re}}$
(
$1.62\times 10^{5}$
). Data from Fage (Reference Fage1936) in the subcritical regime (
${\textit{Re}}=1.1\times 10^{5}$
) are also presented for reference, where the present study again shows good agreement. Figure 9(a) also presents data from the present research at
${\textit{Re}}=2.4\times 10^{5}$
and numerical (i.e. LES) data from Muto et al. (Reference Muto, Tsubokura and Oshima2012) at the same
${\textit{Re}}$
. Again, good qualitative agreement of the
$C_{\!P}$
profile is shown, with base pressure matching almost exactly between the two studies. A slight divergence in
$C_{\!P}$
is observed between
$\phi =90^\circ$
and
$\phi =140^\circ$
. Given that the numerical simulation from Muto et al. (Reference Muto, Tsubokura and Oshima2012) had zero free stream turbulence intensity and a perfectly smooth sphere surface, some difference in the boundary layer separation angle is expected, which likely explains the difference in this region.
Figure 9(b) presents a comparison of the
$\overline {C}_{\!P}$
distribution for a rotating smooth sphere across the
$YZ$
(equatorial) plane between the present research at
${\textit{Re}}=2.3\times 10^{5}$
and
$\alpha =0.1$
, and numerical data from Muto et al. (Reference Muto, Tsubokura and Oshima2012) at
${\textit{Re}}=2.0\times 10^{5}$
and
$\alpha =0.2$
. Given the lack of previous experimental pressure data for rotating spheres and minimal numerical data, this was deemed the best available comparison for validation. Generally, qualitative agreement is demonstrated between the two studies; the base pressure and peak suction on the advancing side are consistent. The present study shows marginally larger peak suction on the retreating side; it is suggested that the difference in
${\textit{Re}}$
and
$\alpha$
between the studies is likely the reason for this offset. The angle at which the ‘kink’ signature of the SV appears in the
$C_{\!P}$
distribution (examined in detail in § 3.2.1) is highly consistent across the studies and agrees to within
$4^\circ$
. This gives credence to the rotating experimental pressure measurements presented later in this study.
Validation of experimental surface pressure measurements against previous studies for (a) the stationary smooth sphere and (b) the rotating smooth sphere.
3.2. Pressure coefficient,
$C_{\!P}$
, distributions
Time-averaged equatorial
$\overline {C}_{\!P}$
distribution at various
${\textit{Re}}$
for the smooth sphere rotating at
$\omega \approx 1$
rev s–1. Advancing side from
$\theta =0\rightarrow 180^\circ$
and retreating side from
$\theta =180\rightarrow 360^\circ$
.
Figure 10 presents equatorial
$\overline {C}_{\!P}$
distributions at various
${\textit{Re}}$
for the smooth sphere at
$\omega \approx 1$
rev s–1 (
$0.02\leqslant \alpha \leqslant 0.11$
). Here
$\theta =0\rightarrow 180^\circ$
marks the retreating side and
$\theta =180\rightarrow 360^\circ$
marks the advancing side. For ease of comparison between boundary layer separation points on the advancing and retreating sides, a new angle is defined.
$\theta _{sep}$
is the azimuthal separation angle relative to the stagnation point; its sign does not change across advancing and retreating sides. The blue (top) distribution in figure 10 shows
$\overline {C}_{\!P}$
for
${\textit{Re}}=0.45\times 10^{5}$
, which was the lowest
${\textit{Re}}$
tested. The boundary layer on the sphere is considered to be wholly laminar on both sides at this
${\textit{Re}}$
, and no subsequent reattachment post separation is observed. On the retreating side, additional momentum is conferred on the boundary layer, hence it can better overcome the adverse pressure gradient and separates later than on the advancing side (Kray, Franke & Frank Reference Kray, Franke and Frank2012). Laminar boundary layer separation (i.e. LS) occurs at
$\theta _{\textit{sep}, \textit{ret}}\approx 95^\circ$
(downstream of sphere shoulder) on the retreating side, versus
$\theta _{\textit{sep}, \textit{ad}v}\approx 80^\circ$
(upstream of sphere shoulder) on the advancing side. Separation angles were visually estimated from the
$C_{\!P}$
distributions. The delayed separation on the retreating side results in lower pressure compared with the advancing side and the sphere experiences the ordinary Magnus effect (Kray et al. Reference Kray, Franke and Frank2012; Kim et al. Reference Kim, Choi, Park and Yoo2014). The trend of
$\theta _{\textit{sep}, \textit{ret}} \gt \theta _{\textit{sep}, \textit{ad}v}$
and lower pressure on the retreating side continues to
${\textit{Re}}\approx 1.3\times 10^5$
, where the separation angles are approximately the same. At
${\textit{Re}}=1.36\times 10^{5}$
in figure 10 the peak suction (
$-\overline {C}_{\! P_{\textit{peak}}}$
) on advancing and retreating sides is approximately equal. This is observed in the orange (second from top)
$C_{\!P}$
distribution where
$-\overline {C}_{\! P_{\textit{peak}}}\approx 0.3$
on both sides. Although not shown here, for
$1.3\times 10^5\lesssim {\textit{Re}}\lesssim 1.5\times 10^{5}$
the separation angle and peak suction were approximately equal on the advancing and retreating sides.
The yellow (third from top) distribution in figure 10 plots
$\overline {C}_{\!P}$
for
${\textit{Re}}=1.94\times 10^5$
. It is observed that
$\theta _{\textit{sep}, \textit{ret}}\approx 110^\circ$
and
$\theta _{\textit{sep}, \textit{ad}v}\approx 120^\circ$
, and that peak suction is larger on the advancing side (
$-\overline {C}_{\!P_{\textit{peak-ad}v}}\approx 0.7$
compared with
$-\overline {C}_{P_{peak-ret}}\approx 0.4$
). The trend of
$\theta _{\textit{sep}, \textit{ad}v} \gt \theta _{\textit{sep}, \textit{ret}}$
, and lower pressure on the advancing side, continues to the highest
${\textit{Re}}$
tested. Kim et al. (Reference Kim, Choi, Park and Yoo2014) conclude that, at a critical
${\textit{Re}}$
and
$\alpha$
, the laminar boundary layer on the advancing side undergoes a transition to turbulence whereas the boundary layer on the retreating side remains laminar. The advancing side transition is attributed to a shear layer instability, of the Kelvin–Helmholtz type, caused by a velocity gradient between fluid moving upstream (due to the non-slip condition at the sphere surface) and fluid moving downstream. Advancing side turbulence results in higher momentum within the boundary layer, which is better able to overcome the adverse pressure gradient. Thus, final TS occurs farther downstream compared with the retreating side (Kim et al. Reference Kim, Choi, Park and Yoo2014). This results in the inverse Magnus effect, with a lift force exerted in the retreating to advancing direction. This phenomenon is best illustrated by figure 11, where, on the advancing side the boundary layer has transitioned to turbulence, as can be seen by the plateau signature of the LSB. On the retreating side, the boundary layer remains laminar and separates earlier.
Time-averaged equatorial
$\overline {C}_{\!P}$
distribution at
${\textit{Re}}=1.76\times 10^{5}$
for the smooth sphere rotating at
$\omega \approx 1$
rev s–1 (
$\alpha =0.02$
).
Time-averaged equatorial peak suction (
$-\overline {C}_{\! P_{\textit{peak}}}$
) at various
${\textit{Re}}$
and
$\omega \approx 1$
rev s–1,
$0.02\leqslant \alpha \leqslant 0.1$
. For (a) smooth and (b) rough spheres, advancing (red
$\triangle$
) and retreating (blue
$\circ$
) sides.
Time-averaged equatorial
$\overline {C}_{\!P}$
distribution at various
${\textit{Re}}$
for the rough sphere rotating at
$\omega \approx 1$
rev s–1. Advancing side from
$\theta =0\rightarrow 180^\circ$
and retreating side from
$\theta =180\rightarrow 360^\circ$
.
Figures 12(a) and 12(b) plot the variation with
${\textit{Re}}$
of peak suction,
$-\overline {C}_{\! P_{\textit{peak}}}$
for the smooth and rough spheres, respectively. The transition from ordinary to inverse Magnus effect for the smooth sphere is seen in figure 12(a), where
$-\overline {C}_{\!P_{\textit{peak-ad}v}}$
becomes larger than
$-\overline {C}_{P_{peak-ret}}$
(crossing of red and blue lines) at
${\textit{Re}}\approx 1.3\times 10^5$
. The transition from ordinary to inverse Magnus effect occurs at a lower
${\textit{Re}}$
for the rough sphere relative to the smooth sphere (at
$0.5 \times 10^{5}\leqslant {\textit{Re}}\leqslant 1\times 10^5$
). No discernible relationship was observed between the variation with
${\textit{Re}}$
of azimuthal angle of peak suction.
Figure 13 presents equatorial
$\overline {C}_{\!P}$
distributions at select
${\textit{Re}}$
for the rough sphere rotating at
$\omega \approx 1$
rev s–1 (
$0.02\leqslant \alpha \leqslant 0.11$
). At
${\textit{Re}}=0.44\times 10^5$
(blue/top profile in figure 13),
$\theta _{\textit{sep}, \textit{ret}}\approx 100^\circ$
and
$\theta _{\textit{sep}, \textit{ad}v}\approx 95^\circ$
, and peak suction is marginally greater on the retreating side. This indicates that the sphere is experiencing the ordinary Magnus effect. For
${\textit{Re}}\gtrsim 0.5\times 10^5$
, separation occurs later on the advancing side, and hence the rough sphere experiences the inverse Magnus effect (up to and including the highest
${\textit{Re}}$
tested). This is compared with
${\textit{Re}}\gtrsim 1.3\times 10^5$
for the onset of inverse Magnus for the smooth sphere. The mechanism explaining the onset of the inverse Magnus effect is the same as for the smooth sphere. The earlier onset of the inverse Magnus effect for the rough sphere is likely due to higher surface roughness causing increased mixing near the surface and resulting in the shear layer instability causing a transition to turbulence at a lower
${\textit{Re}}$
.
Figure 14 presents equatorial
$\overline {C}_{\!P}$
distributions at various
${\textit{Re}}$
for the smooth sphere rotating at
$\omega \approx 5$
rev s–1 (
$0.07\leqslant \alpha \leqslant 0.45$
). When compared with figure 10 at
$\omega \approx 1$
rev s–1, the
$\overline {C}_{\!P}$
distributions at
$\omega \approx 5$
rev s–1 appear smoother, with noticeably fewer fluctuations around the separation points and in the base pressure region. This is a consequence of averaging over a larger number of revolutions in a given time because of higher rotational speed. For a given
${\textit{Re}}$
in all flow regimes, it is observed that higher rotation rates result in a larger peak suction on the retreating side. In the critical and supercritical flow regimes, higher rotation rates result in a larger peak suction on the advancing and retreating sides. Comparing
$\overline {C}_{\!P}$
distributions at
${\textit{Re}}=1.36\times 10^{5}$
(orange/second from top) in figures 10 and 14 shows a 94 % increase in retreating side peak suction from
$\omega \approx 1$
to
$5$
rev s–1 (
$\alpha \approx 0.03$
to
$0.16$
). This is likely due to a relatively larger amount of momentum being conferred on the laminar boundary layer at the surface of the sphere by the higher speed compared with the slower speed sphere. This causes delayed transition to the inverse Magnus effect; table 4 gives the
${\textit{Re}}$
at which the onset of inverse Magnus occurs at the various test conditions. For the smooth sphere, increasing non-dimensional rotation rate leads to a delayed onset of inverse Magnus with respect to
${\textit{Re}}$
. However, for the rough sphere at
$\omega \approx 5$
rev s–1, the
${\textit{Re}}$
at which the flow transitions to inverse Magnus is lower than for
$\omega \approx 3$
rev s–1. It is suggested that the interplay between rotational speed and surface roughness makes the onset of shear layer instability on the advancing side hard to predict. Kim et al. (Reference Kim, Choi, Park and Yoo2014) derived empirical formulae to predict the location of flow separation as a function of
${\textit{Re}}$
and
$\alpha$
, and measured lift and drag forces for a smooth rotating sphere. The novelty of this study lies in the direct experimental collection and analysis of surface pressure data for rotating spheres. Agreement was found with Kim et al. (Reference Kim, Choi, Park and Yoo2014) on the mechanism causing the inverse Magnus effect. The present research builds on previous studies by quantifying the effect of surface roughness on the transition from ordinary to inverse Magnus. An earlier onset of the inverse Magnus effect with respect to
${\textit{Re}}$
for the rough sphere was demonstrated in comparison with the smooth sphere.
Time-averaged equatorial
$\overline {C}_{\!P}$
distribution at various
${\textit{Re}}$
for the smooth sphere rotating at
$\omega \approx 5$
rev s–1. Advancing side from
$\theta =0\rightarrow 180^\circ$
and retreating side from
$\theta =180\rightarrow 360^\circ$
.
Approximate
${\textit{Re}}$
at which transition from ordinary to inverse Magnus effect occurs.
3.2.1. Laminar separation bubble and SV
For the smooth sphere rotating at the lowest rotation rates (
$\omega \approx 1$
rev s–1), the first certain appearance of the SV and LSB in the flow is observed at
${\textit{Re}}=1.76\times 10^{5}$
, see figure 11. On the advancing side, LS occurs at
$\theta _{\textit{ad}v\text{-}\textit{LS}}\approx 87^\circ$
and a clear plateau at
$\overline {C}_{\!P}=-0.4$
is observed which straddles the shoulder. A kink is observed at
$\theta _{\textit{ad}v}=100^\circ$
, consistent with the signature of the SV. It is observed that during the transition from ordinary to inverse Magnus effect (see figure 12
a,
$1.25\times 10^{5}\leqslant {\textit{Re}}\leqslant 1.55\times 10^{5}$
), when peak suction is approximately equal on advancing and retreating sides, the SV and LSB are not seen in the
$\overline {C}_{\!P}$
distributions, but are apparent after the transition to inverse Magnus. For the smooth sphere at
$\omega \approx 1$
rev s–1 in figure 10, the signatures of the SV and LSB are observed to exist in the advancing side
$C_{\!P}$
distributions up to and including the highest
${\textit{Re}}$
tested. The equatorial
$\overline {C}_{\!P}$
signature of the SV and LSB appears to change in the high-critical/supercritical regime; the combined imprint is observed to transition from a plateau-kink into a w-like shape. An example of the w imprint can be found in the advancing side
$\overline {C}_{\!P}$
distribution at
${\textit{Re}}=3.03\times 10^{5}$
,
$\theta \approx 95^\circ$
(green/bottom profile in figure 10). This suggests the SV increases in strength (relative to the LSB) as
${\textit{Re}}$
increases. A similar change is seen in the
$C_{\!P}$
distributions from Chopra & Mittal (Reference Chopra and Mittal2022) for flow past a cylinder, where the SV causes a sharper kink as
${\textit{Re}}$
increases, indicating an increase in strength. Chopra & Mittal (Reference Chopra and Mittal2022) used sign changes in the surface skin friction coefficient,
$C_{\!f}$
, distribution to identify the separation and attachment points of the SV and LSB; these points were then mapped onto
$C_{\!P}$
distributions. Figure 15 presents the variation of separation and attachment angles with
${\textit{Re}}$
on the advancing side of the smooth sphere rotating at
$\omega \approx 1$
rev s–1. It should be noted that separation and attachment angles were qualitatively identified based on the
$\overline {C}_{\!P}$
distribution, with reference to Chopra & Mittal (Reference Chopra and Mittal2022), hence a degree of uncertainty is present in the results. The SS and SA were not able to be identified from the
$C_{\!P}$
distribution when the flow was devoid of an LSB at
${\textit{Re}}\lt 1.76\times 10^{5}$
. Generally, figure 15 provides good qualitative agreement with the results from Chopra & Mittal (Reference Chopra and Mittal2022) for flow past a cylinder, and Parekh et al. (Reference Parekh, Chaplot and Mittal2024) for flow past the non-seam side of a cricket ball.
Variation of separation and attachment angles with
${\textit{Re}}$
on the advancing side of the smooth sphere rotating at
$\omega \approx 1$
rev s–1, SV and LSB regions are shaded in pink and blue, respectively.
For the smooth sphere at
$\omega \approx 1$
rev s–1, the signature of the LSB is also observed on the retreating side for
$1.94\times 10^{5}\leqslant {\textit{Re}}\leqslant 2.44\times 10^{5}$
; the
$\overline {C}_{\!P}$
distribution at
${\textit{Re}}=1.94\times 10^{5}$
in figure 10 (yellow/third from top) provides an example. Laminar separation occurs at approximately the same point on advancing and retreating sides (
$\theta _{LS}\approx 90^\circ$
), however, TS occurs later on the advancing side (
$\theta _{\textit{ad}v\text{-}\textit{TS}}\approx 120^\circ$
versus
$\theta _{\textit{ret-TS}}\approx 110^\circ$
). This suggests that the turbulent boundary layer on the advancing side is of higher momentum than the turbulent boundary layer on the retreating side and hence separates later. After LS, the separated shear layer experiences significant mixing due to eddies generated by a Kelvin–Helmholtz type instability (Singh & Mittal Reference Singh and Mittal2005). This mixing energises the boundary layer and results in reattachment. It is conjectured that the increased momentum of the advancing side turbulent boundary layer, and hence its later separation, is due to more intense mixing and entrainment of backflow on the advancing side. It is likely that larger eddies are generated on the advancing side due to the relative opposition in movement between the free stream flow and surface compared with the retreating side. The presumed larger eddies are expected to lead to more intense mixing, a higher momentum turbulent boundary layer and hence later separation on the advancing side. Further evidence is required to confirm this conjecture.
It is observed from figures 10 and 15 that as
${\textit{Re}}$
increases, LS occurs farther downstream on the advancing side,
$\theta _{\textit{ad}v\text{-}\textit{LS}}\approx 74^\circ$
at
${\textit{Re}}=0.45\times 10^{5}$
compared with
$\theta _{\textit{ad}v\text{-}\textit{LS}}\approx 98^\circ$
at
${\textit{Re}}=3.03\times 10^{5}$
. The same trend is observed for the rough sphere (see figure 13). In the flow past the rough sphere at
$\omega \approx 1$
rev s–1, the signatures of the SV and LSB in the
$\overline {C}_{\!P}$
distribution disappear after
${\textit{Re}}\approx 2\times 10^{5}$
(see green/bottom line in figure 13). It is interpreted that the combination of surface roughness and high
${\textit{Re}}$
causes intense mixing in the flow closest to the surface to such an extent that it leads to the annihilation of the SV and LSB. A similar phenomenon was observed by Parekh et al. (Reference Parekh, Chaplot and Mittal2024) in the flow past a cricket ball, which supports the interpretation of the present study. On the (smooth) non-seam side, the SV was observed to exist up to the highest
${\textit{Re}}$
tested, however, on the (rough) seam side, the SV disappeared in the flow above a certain
${\textit{Re}}$
. Parekh et al. (Reference Parekh, Chaplot and Mittal2024) attributed this disappearance to increased mixing in the near-surface flow due to the seam of the ball which acts as a large roughness element.
Equatorial
$\overline {C}_{\!P}$
distribution at
${\textit{Re}}=1.95\times 10^{5}$
for the smooth sphere rotating at
$\omega \approx 5$
rev s–1 (
$\alpha =0.11$
), showing distributions time-averaged over
$N=3$
,
$10$
and
$65$
revolutions.
Comparing the
$\overline {C}_{\!P}$
imprints of the SV and LSB in the flow past the smooth sphere at
$\omega \approx 1$
and
$5$
rev s–1 (figures 10 and 14), it is observed that, at higher rotation rates, the SV causes a more significant kink in the
$\overline {C}_{\!P}$
distribution across the
${\textit{Re}}$
range. Given the increased smoothening at higher rotation rates discussed earlier, the
$\overline {C}_{\!P}$
distribution at
${\textit{Re}}=1.95\times 10^{5}$
,
$\omega \approx 5$
rev s–1 was time-averaged across
$N=3$
,
$10$
and
$65$
revolutions. Figure 16 shows that time-averaging over a larger number of revolutions significantly smooths fluctuations in the base pressure region, however, has a minimal effect on
$\overline {C}_{\!P}$
distribution in the shoulder region of the advancing side (especially between
$N=10$
and
$65$
). This indicates that the observed difference in SV–LSB signature at
$\omega \approx 5$
rev s–1 is not a consequence of time-averaging. Hence, evidence is found that increasing
$\alpha$
leads to a stronger SV relative to the LSB, and the combined signature is modified to include a larger dip/kink.
Time-averaged equatorial
$\overline {C}_{\!P}$
distribution at various
${\textit{Re}}$
at two fixed non-dimensional rotation rates,
$\alpha$
, for the smooth sphere. Advancing side from
$\theta =0\rightarrow 180^\circ$
and retreating side from
$\theta =180\rightarrow 360^\circ$
.
3.2.2. Pressure coefficient,
$C_{\!P}$
, distributions at constant non-dimensional rotation rates,
$\alpha$
The experimental design in the present study primarily used fixed rotational speeds (
$\omega$
) rather than fixed non-dimensional rotation rates (
$\alpha$
). To interpret trends in the variation of
$\overline {C}_{\!P}$
distribution with
${\textit{Re}}$
, to isolate the effects of rotation, and to support the findings at fixed rotational speeds in § 3.2, a selection of data are provided for the smooth sphere at fixed
$\alpha$
values in figure 17. Figures 17(a) and 17(b) present the variation of
$\overline {C}_{\!P}$
with
${\textit{Re}}$
for
$\alpha \approx 0.05$
and
$0.1$
, respectively. Figure 17(c) shows
$\overline {C}_{\!P}$
distributions at
${\textit{Re}}=1.9$
and
$2.3\times 10^{5}$
, each at
$\alpha \approx 0.05$
and
$0.1$
, allowing for ease of comparison.
Figure 17(b) shows that for
$\alpha \approx 0.1$
the transition from ordinary to inverse Magnus effect occurs at
${\textit{Re}}=1.33\times 10^{5}$
. This transitional
${\textit{Re}}$
is consistent with table 4. For
$\alpha \approx 0.1$
, at the transitional
${\textit{Re}}$
,
$-\overline {C}_{\! P_{\textit{peak}}}=0.31$
on advancing and retreating sides (see yellow line in figure 17
b). The value of peak suction at the transitional
${\textit{Re}}$
also agrees with the fixed
$\omega$
data shown in figure 10 at
$\omega =1$
rev s–1, where
$-\overline {C}_{\! P_{\textit{peak}}}\approx 0.3$
at the transition from ordinary to inverse Magnus. As found in § 3.2, for a given
${\textit{Re}}$
in the critical or supercritical regime, higher rotation rates result in larger peak suction on both the advancing and retreating sides. This is elucidated by figure 17(c) where increases in advancing and retreating side
$-\overline {C}_{\! P_{\textit{peak}}}$
are seen for both
${\textit{Re}}=1.9\times 10^{5}$
and
${\textit{Re}}=2.3\times 10^{5}$
, as rotation rate increases by a factor of two from
$\alpha \approx 0.05$
to
$\alpha \approx 0.1$
.
Compared with figures 10 and 14, figure 17(b) allows for the effects of increasing
${\textit{Re}}$
to be isolated from the effects of variation in non-dimensional rotation rate. In figure 17(b), it is observed that as
${\textit{Re}}$
increases, the kink signature of the SV on the retreating side transitions to a w-like imprint (see light blue line in figure 17
b). This provides more certainty that the changing SV signature is attributed to increasing
${\textit{Re}}$
and supports the findings of Chopra & Mittal (Reference Chopra and Mittal2022) that increasing
${\textit{Re}}$
leads to a sharper SV kink, indicating an increase in the strength of the SV. In agreement with the constant rotational speed data, figure 17(c) shows that, for fixed
${\textit{Re}}$
, increases in rotation rate result in more significant and deeper kinks associated with the SV (observed in the
$250^\circ \lesssim \theta \lesssim 270^\circ$
region). This provides evidence that increasing
$\alpha$
also leads to a stronger SV relative to the LSB.
3.3. Cross-flow PIV and WTV
Graftieaux et al. (Reference Graftieaux, Michard and Grosjean2001) developed a vortex identification algorithm combining proper orthogonal decomposition with two new vortex identification functions,
$\varGamma _{1}$
and
$\varGamma _{2}$
. The
$\varGamma _{1}$
dimensionless scalar may be used to identify the locations of vortex centres. They state that near a vortex centre,
$\lvert \varGamma _{1}\rvert$
reaches values between
$0.9$
and
$1.0$
. For simplicity, in the present research, the maximum and minimum values of
$\varGamma _{1}$
have been used to identify the centres of clockwise (CW) and anticlockwise (ACW) rotating vortices, respectively. Graftieaux et al. (Reference Graftieaux, Michard and Grosjean2001) propose that vortex boundaries can be identified using the following condition:
$\lvert \varGamma _{2}\rvert \gt 2/\pi$
, which corresponds to the flow being locally dominated by rotation. However, they state that the precise relationship between
$\varGamma _{2}$
, local flow state, and finite element area is not yet fully understood. In this study, various
$\varGamma _{2}$
thresholds were investigated with
$\lvert \varGamma _{2}\rvert \gt 0.715$
found to provide the best visual match to the velocity fields and apparent vortex extents; this was subsequently used throughout the analysis. A vortex identification method implementing the algorithm by Graftieaux et al. (Reference Graftieaux, Michard and Grosjean2001) has been modified from code written by Endrikat (Reference Endrikat2024).
Flow past the stationary smooth sphere (
$\alpha =0$
) at
${\textit{Re}}=0.5\times 10^{5}$
and
$y/D=1.5$
: (a) velocity field and (b) contour of time-averaged streamwise non-dimensional vorticity component (
$\omega _{y}^{*}$
).
Cross-flow PIV was used in this study to investigate flow phenomena in the wake of the stationary smooth sphere. Figure 18 presents the velocity field and contour plot of streamwise non-dimensional vorticity (
$\omega _{y}^{*}$
) at
${\textit{Re}}=0.5\times 10^{5}$
for the stationary sphere; velocity has been time-averaged across 500 image-pairs, where
$\omega _{y}^{*}=\omega _{y}(D/{U_{\infty }})$
, and
$\omega _{y}$
is the streamwise component of vorticity. Figure 18(a) shows the centre locations of CW and ACW rotating vortices, identified using the
$\varGamma _{1}$
scalar. The centre locations are approximately symmetrical across the
$XY$
plane at
$z=0$
. The vortices formed in the wake of the stationary sphere – seen in figure 18(a) – are interpreted as part of a horseshoe vortex system, often studied in the context of leading-edge regions of endwall juncture flows in turbomachinery (Praisner & Smith Reference Praisner and Smith2006). In the present study, the shaft results in the formation of pressure gradients in the surrounding flow. When the boundary layer on the sphere, upstream of the shaft, encounters an adverse pressure gradient, it will undergo three-dimensional separation (Baker Reference Baker1979). The separated shear layer rolls up into a system of vortices and is swept around the base of the shaft to form the characteristic horseshoe vortex shape (Baker Reference Baker1979). It is theorised that the horseshoe vortex system is distorted and the vortex centres pulled towards the centre of the sphere through three-dimensional interaction with the sphere wake and associated pressure gradients (as can be seen in figure 18
a). The present research does not explore the three-dimensional horseshoe vortex phenomena. Figure 18(b) presents a contour plot of the streamwise component of non-dimensional vorticity (
$\omega _{y}^{*}$
) at the same test condition. It is observed that vortices, as identified by
$\lvert \varGamma _{1}\rvert _{max}$
(red and blue crosses), are not in good agreement with the areas of high and low vorticity. Local quantities, such as vorticity, are highly intermittent in relation to vortex identification, due to the presence of turbulence on a grid-scale significantly smaller than that of the PIV measurements (Graftieaux et al. Reference Graftieaux, Michard and Grosjean2001). Vorticity was computed using MATLAB’s numerical curl function, which implements the central differencing scheme to calculate velocity derivatives. This second-order accurate scheme amplifies noise created through time-averaging of small-scale turbulent fluctuations, hence the non-dimensional vorticity field appears noisy and is somewhat ineffective for vortex identification. However, as shown below, the
$\varGamma _{2}$
scalar performs vortex identification highly effectively in this context due to its inherent Galilean invariance.
Use of
$\varGamma _{1}$
and
$\varGamma _{2}$
scalar quantities to identify WTV at
$y/D=1.5$
,
${\textit{Re}}=0.5\times 10^5$
and
$\alpha =0.45$
(
$\omega \approx 5$
rev s–1), data time-averaged from 60 image-pairs. Contours of (a)
$\varGamma _{1}$
and (b)
$\varGamma _{2}$
scalars, and (c) velocity field with identified vortex centres and boundaries.
Contour of
$\varGamma _{2}$
at
$y/D=2.0$
,
${\textit{Re}}=0.5\times 10^5$
and
$\alpha =0.45$
(
$\omega \approx 5$
rev s–1), data time-averaged from 60 image-pairs.
Figure 19 demonstrates the process by which vortices were identified, illustrated for
${\textit{Re}}=0.5\times 10^{5}$
,
$\alpha \approx 0.45$
, at
$y/D=1.5$
, and time-averaged across 60 image-pairs. Figure 19(a) shows the
$\varGamma _{1}$
field, where a clear maximum and minimum are observed which correspond to the centres of vortices. Figure 19(c) shows that the identified vortex centres are in very good visual agreement with the velocity field. Figure 19(b) presents a contour of
$\varGamma _{2}$
where two distinct areas of large
$\lvert \varGamma _{2}\rvert$
are observed. The vortices associated with these two regions of large
$\lvert \varGamma _{2}\rvert$
are interpreted as counter-rotating WTV, as observed experimentally by Li et al. (Reference Li, Zhang, Liu and Gao2023b
) in the wake of a rotating sphere, and computationally by Parekh et al. (Reference Parekh, Chaplot and Mittal2024) in the wake of a stationary cricket ball. Wing-tip-like vortices – as coined by Parekh et al. (Reference Parekh, Chaplot and Mittal2024) – are formed by flow leakage, in the polar regions, from the advancing side (relatively higher pressure) to the retreating side (relatively lower pressure) due to the pressure differential which exists when the sphere is experiencing the ordinary Magnus effect. In the quadrant where
$x\gt 0$
and
$z\lt 0$
, an area of large
$\lvert \varGamma _{2}\rvert$
is observed which does not exist on the non-shaft side. This region of high
$\varGamma _{2}$
, alongside the circular region of low
$\varGamma _{2}$
at (
$x\approx 70$
mm,
$z\approx 50$
mm), is interpreted as the signature of the horseshoe vortex system discussed earlier for the stationary sphere. On the advancing side (top-half of figure 19
b), the horseshoe vortex is rotating counter to the WTV; this results in the horseshoe vortex being pushed radially outwards by the flow-leakage from the retreating to the advancing side. Conversely, on the retreating side (bottom-half of figure 19
b) the horseshoe vortex is corotating with the WTV. It is well established in the literature that corotating vortices eventually merge (Cerretelli & Williamson Reference Cerretelli and Williamson2003). On the retreating side at
$y/D=1.5$
, nascent merging of the horseshoe vortex and CW rotating WTV can be seen in figure 19(b), where two semi-distinct areas of large
$\lvert \varGamma _{2}\rvert$
are beginning to coalesce. The evolution of the vortex merging can be seen in figure 20, where the two vortices are no longer distinct at
$y/D=2.0$
and appear as a contiguous region of high
$\lvert \varGamma _{2}\rvert$
. The threshold for vortex boundary identification (
$\lvert \varGamma _{2}\rvert \gt 0.715$
) was chosen in order to distinguish the horseshoe vortex from the WTV at
$y/D=1.5$
.
Figures 21(a) and 21(b) compare time-averaged velocity fields for 60 and 500 image-pair averages, respectively, at
${\textit{Re}}=0.5\times 10^{5}$
,
$\alpha =0.45$
and
$y/D=1.5$
. Figure 21(b
) shows that the 500 image-pair average significantly smooths the flow field and WTVs appear smeared with less distinct swirl compared with the 60 image-pair average. This smearing is a consequence of WTVs spatially wandering/oscillating during a PIV test run. In an effort to qualitatively understand the phenomenon of WTV wander, raw image data from PIV test runs of 500 image pairs were divided into 10 subsets of 50 image pairs and then averaged. The centres and boundaries of the WTVs were then computed for each averaged subset using
$\varGamma _{1}$
and
$\varGamma _{2}$
fields, respectively. Figure 22 shows the variation of WTV centres and extents at
${\textit{Re}}=0.5\times 10^5$
,
$\alpha =0.45$
and
$y/D=1.5$
. This provides further evidence of the spatial oscillation of the counter-rotating vortex pair. The frequency of WTV oscillation was not able to be deduced using the data collected.
Velocity fields on the
$XZ$
plane at
$y/D=1.5$
for
${\textit{Re}}=0.5\times 10^{5}$
and
$\alpha =0.45$
(
$\omega \approx 5$
rev s–1), time-averaged from (a) 60 and (b) 500 image-pairs.
Wing-tip-like vortex wander:
$y/D=1.5$
,
${\textit{Re}}=0.5\times 10^5$
and
$\alpha =0.45$
(
$\omega \approx 5$
rev s–1), for 10 averaged subsets of 50 image-pairs. (a) Vortex centres identified using the
$\varGamma _{1}$
dimensionless scalar and (b) overlaid (layered) vortex areas identified using the
$\varGamma _{2}$
scalar.
Velocity fields, time-averaged from 500 image-pairs, for
${\textit{Re}}=0.5\times 10^{5}$
and
$\alpha =0.45$
(
$\omega \approx 5$
rev s–1), at (a)
$y/D=1.5$
and (b)
$y/D=2.0$
; (c) comparison of vortex centres at
$y/D=1.5$
and
$2.0$
from respective 500 image-pair runs divided into 10 subsets of 50 image-pairs for analysis.
Figure 22 shows that the centre of the CW rotating WTV is consistently further towards the retreating (bottom) side than the centre of the ACW rotating WTV, with a mean of
$\overline {z}_{CW-cen}=8$
mm versus
$\overline {z}_{ACW-cen}=48$
mm. Li et al. (Reference Li, Zhang, Liu and Gao2023b
) also observed an offset between vortex centres for CW and ACW rotating WTVs; they referred to this phenomenon as a tilted vortex pair. In their study, they used a rotating support shaft and suggested that tilting may be caused by the flow leakage associated with the shaft under rotation. In the present work, a stationary sleeve surrounded the rotating shaft. Hence, it is instead proposed that the tilted vortex pair is caused by the merging of the retreating side horseshoe vortex generated due to the presence of the support shaft and CW rotating WTV.
Figure 23 compares the time-averaged velocity fields and WTV centres at two downstream planes,
$y/D=1.5$
(figure 23
a) and
$y/D=2.0$
(figure 23
b), for
${\textit{Re}}=0.5\times 10^{5}$
and
$\alpha =0.45$
. From figure 23(c), it is observed that the mean centre of the ACW rotating WTV changes relatively little between planes,
$(x=-35,z=48)$
mm at
$y/D=1.5$
versus
$(x=-40,z=35)$
mm at
$y/D=2.0$
. The CW rotating WTV mean centre shows a larger change, however,
$(x=44,z=8)$
mm at
$y/D=1.5$
versus
$(x=70,z=-28)$
mm at
$y/D=2.0$
. Two remarks are made based on these observations. Firstly, given the relative stasis of the ACW WTV centre, it is surmised that the vortex core line is approximately parallel to the free stream flow. Secondly, the shift in CW WTV centre position between the two downstream planes is due to the progress of vortex merging between the horseshoe and WTV (visualised by comparison of figures 19
b and 20).
Contours of time-averaged streamwise non-dimensional vorticity component (
$\omega _{y}^{*}$
) at
$y/D=1.5$
, for (a)
${\textit{Re}}=0.5\times 10^{5}$
,
$\alpha \approx 0.3$
and (b)
${\textit{Re}}=0.7\times 10^{5}$
,
$\alpha \approx 0.3$
. (c) Comparison of vortex centres at
${\textit{Re}}=0.5\times 10^5$
and
$0.7\times 10^5$
from respective 500 image-pair runs divided into 10 subsets of 50 image-pairs.
Figure 24 presents time-averaged non-dimensional vorticity (
$\omega _{y}^{*}$
) contours at
$y/D=1.5$
for
${\textit{Re}}=0.5\times 10^5$
and
$0.7\times 10^5$
. It is noted that stainless steel was left exposed at the joint between the sphere and the shaft on the rig, causing reflection of the laser sheet. This resulted in spurious velocity measurements, hence the non-dimensional vorticity calculated in this region does not represent the physical flow. For approximately the same
$\alpha$
, figures 24(a) and 24(b) compare the effect of increasing
${\textit{Re}}$
. The
$\lvert \omega _{y}^{*}\rvert _{max}$
increases by
$22.6\,\%$
from
${\textit{Re}}=0.5\times 10^5$
to
${\textit{Re}}=0.7\times 10^5$
, suggesting that higher
${\textit{Re}}$
likely result in stronger WTVs. This is given credence by figure 25, which shows the pressure difference between advancing and retreating sides increases from
$\Delta \overline {C}_P=0.47$
at
${\textit{Re}}\approx 0.5\times 10^{5}$
to
$\Delta \overline {C}_P=0.53$
at
${\textit{Re}}\approx 1\times 10^{5}$
(at similar
$\alpha$
). Hence, evidence is found that WTV strength depends on the pressure difference between advancing and retreating sides (governed by
${\textit{Re}}$
and
$\alpha$
); as found by Parekh et al. (Reference Parekh, Chaplot and Mittal2024). Figure 24(c) compares WTV centres for
${\textit{Re}}=0.5\times 10^5$
and
$0.7\times 10^5$
computed using
$\varGamma _{1}$
. It is observed that increasing
${\textit{Re}}$
whilst maintaining the same
$\alpha$
does not significantly alter the WTV centre position.
Time-averaged equatorial
$\overline {C}_{\!P}$
distribution at
${\textit{Re}}=0.46\times 10^5$
(red) and
${\textit{Re}}=0.96\times 10^5$
(blue) for the smooth sphere with
$0.22\leqslant \alpha \leqslant 0.27$
.
Velocity fields (i) and equatorial
$C_{\!P}$
distributions (ii) at varying
${\textit{Re}}$
and constant non-dimensional rotation rate (
$\alpha \approx 0.28$
) at
$y/D=1.5$
, all averaged from 500 image pairs: (a)
${\textit{Re}}=0.5\times 10^{5}$
, (b)
$1.15\times 10^{5}$
, (c)
$1.74\times 10^{5}$
.
Parekh et al. (Reference Parekh, Chaplot and Mittal2024) performed LES of the flow past a static cricket ball. They observed a pair of counter-rotating WTVs caused by flow leakage in the polar regions due to a pressure difference between the seam and the non-seam sides. Parekh et al. (Reference Parekh, Chaplot and Mittal2024) also detected a change in the WTV polarity when the peak suction on the non-seam side became larger than on the seam side as
${\textit{Re}}$
was increased. They coined these RWTVs. Changes in WTV polarity were not experimentally observed by Li et al. (Reference Li, Zhang, Liu and Gao2023b
) as they tested at a single
${\textit{Re}}$
. As well as a large ‘primary’ pair of counter-rotating vortices, Parekh et al. (Reference Parekh, Chaplot and Mittal2024) detected smaller secondary and tertiary WTV pairs. It is noted that multiple pairs of counter-rotating vortices were not detected in this study.
Figure 26 presents velocity fields and corresponding equatorial
$\overline {C}_{\!P}$
distributions for
${\textit{Re}}=0.5\times 10^{5}$
,
$1.15\times 10^{5}$
and
$1.74\times 10^{5}$
(figure 26
a to figure 26
c); all tests were performed at the same non-dimensional rotation rate,
$\alpha \approx 0.28$
, and the flow fields were obtained by averaging across 500 image pairs. Figure 26(a) gives the velocity field and
$\overline {C}_{\!P}$
distribution at
${\textit{Re}}=0.5\times 10^5$
. The sphere is experiencing the ordinary Magnus effect at this
${\textit{Re}}$
; figure 26(a ii) shows the pressure on the retreating side is lower than that of the advancing side, hence flow leakage occurs in the polar regions from the advancing to the retreating side. The resultant WTV formulation has been discussed in detail in the preceding analysis. Figure 26(b) shows the velocity field and
$\overline {C}_{\!P}$
distribution close to the point of transition from ordinary to inverse Magnus effect; for
$\alpha \approx 0.28$
this occurs at
${\textit{Re}}=1.15\times 10^{5}$
. The pressure on advancing and retreating sides is approximately equal,
$\overline {C}_{\!P}\approx -0.38$
. The flow field shows that a counter-rotating vortex pair is not formed in the wake under these conditions; flow leakage does not occur in the polar regions due to the lack of a pressure gradient between advancing and retreating sides. This mode therefore resembles the stationary sphere (figure 18) where the horseshoe vortex system is the dominant wake structure. Figure 26(c) shows the flow field and
$\overline {C}_{\!P}$
distribution at
${\textit{Re}}=1.74\times 10^{5}$
when the sphere is experiencing the inverse Magnus effect (
$\overline {C}_{\!P}$
is lower on the advancing side). The mechanism for the transition from ordinary to inverse Magnus is discussed in § 3.2. In this mode, flow leakage occurs from the higher-pressure retreating side to the lower-pressure advancing side and the polarity of the counter-rotating vortex pair switches to form a RWTV pair. Comparing figures 26(a) and 26(c), it is observed that RWTVs are ‘tilted’ in a different orientation to that of the WTVs; the vortex centres are in markedly different positions. This is explained by the merging of the shaft side WTV with either the retreating side (
$z\lt 0$
) horseshoe vortex which rotates CW or the advancing side (
$z\gt 0$
) horseshoe vortex which rotates ACW. When the sphere is experiencing the ordinary Magnus effect (figure 26
a) the shaft side WTV rotates CW and hence merges with the like-sign retreating side horseshoe vortex. This vortex merging results in a counter-rotating vortex pair tilted in the same orientation as observed by Li et al. (Reference Li, Zhang, Liu and Gao2023b
), who only observed the ordinary Magnus effect in their study. After transition to inverse Magnus as
${\textit{Re}}$
increases, the counter-rotating vortex pair switches polarity. The shaft side WTV rotates ACW and hence no longer merges with the retreating side horseshoe vortex but with the advancing side horseshoe vortex instead. Hence, the WTV tilt orientation changes upon transition from ordinary to inverse Magnus due to reversal in vortex sign and consequent changes in vortex merging. The progress of vortex merging between shaft side RWTV and advancing side horseshoe vortex is observed between the planes
$y/D =1.5$
and
$y/D=2.0$
, in the same manner discussed previously for WTVs. Figure 27 shows contours of
$\varGamma _{2}$
for both planes; at
$y/D=1.5$
the horseshoe vortex and WTV are distinct. By
$y/D=2.0$
, a homogeneous vortex has formed with the vortex centre shifting towards the advancing side. The conclusions of the analysis presented in preceding paragraphs relating to the counter-rotating vortex pair formed under the ordinary Magnus effect (i.e WTVs) also apply to the counter-rotating vortex pair formed under the inverse Magnus effect (i.e. RWTVs) and are not presented in duplicate.
The combination of surface pressure data and flow fields measured using PIV has enabled, to the best of the authors’ knowledge, the first experimental observation of reversal in counter-rotating vortex polarity in the wake of a rotating sphere due to the shifting pressure gradient between advancing and retreating sides caused by the transition from ordinary to inverse Magnus effect.
Contours of
$\varGamma _{2}$
in the wake of the smooth sphere at
$\alpha =0.28$
(
$\omega \approx 11$
rev s–1) and
${\textit{Re}}=1.74\times 10^{5}$
averaged from 500 image pairs for (a)
$y/D=1.5$
and (b)
$y/D=2.0$
.
4. Conclusions
This paper has examined in detail the flow past both stationary and rotating spheres of different surface roughness for experimental conditions of
$0.5\times 10^{5}\leqslant {\textit{Re}}\leqslant 3\times 10^{5}$
. This has provided the first direct experimental evidence for the coexistence of both a LSB and a SV on the advancing side of a rotating sphere when the inverse Magnus effect was experienced. Rotating tests were undertaken at three rotational speeds,
$\omega \approx 1$
,
$3$
and
$5$
rev s–1, and spanned the non-dimensional rotation rate range
$0.02\leqslant \alpha \leqslant 0.45$
. The spheres were rotated via a shaft oriented perpendicular to the free stream flow. Surface pressure data were collected for the hemisphere on the non-shaft side, using a rotating spline of 20 pressure taps spanning from equator to pole. Vortex structures in the wake of the smooth rotating sphere were investigated for
$0.5\times 10^5\leqslant {\textit{Re}}\leqslant 1.74\times 10^5$
using cross-flow at two downstream planes.
Time-averaged pressure coefficient (
$\overline {C}_{\!P}$
) distributions on the
$YZ$
(equatorial) plane were plotted for rotating smooth and rough spheres. The
$\overline {C}_{\!P}$
signatures of the LSB and SV were observed on the advancing side of both spheres when the inverse Magnus effect was experienced. For the smooth sphere at the highest
${\textit{Re}}$
, the combined plateau-kink imprint of the LSB and SV was observed to transition to a w-like signature, which indicated the SV increased in strength (relative to the LSB) as
${\textit{Re}}$
increased. For the smooth sphere, an increased rotation rate (
$\alpha$
) was found to lead to a larger kink in the
$\overline {C}_{\!P}$
distribution. This indicated that increased
$\alpha$
resulted in an SV of higher strength relative to the LSB. Variation of peak suction with
${\textit{Re}}$
on the advancing and retreating sides again showed that the transition to inverse Magnus occurred at a lower
${\textit{Re}}$
for the rough sphere compared with the smooth. It was suggested that roughness caused increased mixing near the surface, which resulted in the Kelvin–Helmholtz instability, and hence the transition to turbulence, occurring at a lower
${\textit{Re}}$
. Imprints of the LSB and SV disappeared in the
$\overline {C}_{\!P}$
distribution for the rough sphere (
$\omega \approx 1$
rev s–1) for
${\textit{Re}}\gtrsim 2\times 10^{5}$
. A combination of increased
${\textit{Re}}$
and surface roughness was thought to increase mixing near the surface to such an extent that the LSB and SV were annihilated.
Velocity fields in the wake of the smooth sphere were measured using PIV with the vortex identification functions
$\varGamma _{1}$
and
$\varGamma _{2}$
used to identify vortex centres and boundaries, respectively. The streamwise component of non-dimensional vorticity was computed using the numerical curl of the velocity field. Non-dimensional vorticity fields generally did not provide information about the centre and extent of vortices. Sub-grid scale turbulent fluctuations in the flow were amplified by numerical differentiation in the curl calculation resulting in high levels of noise in non-dimensional vorticity fields. In the wake of the stationary smooth sphere, a horseshoe vortex system was identified on the shaft side. In the wake of the rotating smooth sphere, a pair of counter-rotating WTVs were detected when the sphere was experiencing the ordinary Magnus effect. The WTVs were generated by flow leakage from the relatively high-pressure advancing side to the relatively low-pressure retreating side. The strength of the WTVs, as measured by maximum streamwise non-dimensional vorticity magnitude, was found to be dependent on the pressure difference between advancing and retreating sides. The non-shaft side WTV was found to have a vortex core line approximately parallel to the free stream flow. Tilting of the WTV pair was found to be caused by the merging of the retreating side horseshoe vortex and CW rotating (shaft side) WTV. During the transition from ordinary to inverse Magnus effect, the equatorial pressure coefficient distribution showed that pressure on advancing and retreating sides was approximately equal. A counter-rotating vortex pair was not detected at this transitional
${\textit{Re}}$
because the pressure gradient across advancing and retreating sides was approximately equal to zero; however, horseshoe vortices were clearly observed. As
${\textit{Re}}$
was increased and the sphere began to experience the inverse Magnus effect, the polarity of the counter-rotating vortex pair reversed to form a RWTV pair. The pressure coefficient distribution showed that pressure was higher on the retreating versus the advancing side. Hence, flow leakage occurred from the retreating to the advancing side and vortex polarity was reversed. The RWTVs were tilted in a different orientation to WTVs due to the merging of the advancing side horseshoe vortex and the ACW rotating (shaft side) RWTV.
In summary, this work represents the first experimental observation of vortex polarity reversal on a sphere subjected to the inverse Magnus effect. The new evidence and insight into the complex fluid dynamics of such structures can inform future exploitation of the Magnus effect.
Acknowledgements
The authors would like to thank C. Bricker for her help in setting up the PIV experiments.
Funding
This research received no specific grant from any funding agency, commercial or not-for-profit sectors.
Declaration of interests
The authors report no conflict of interest.
Author contributions
L.G.M. conducted the experiments and analysis, prepared the figures and wrote the original draft. J.A.S. designed the experimental rig and methodology, reviewed and edited the paper and supervised the project.
Open access
