4.1. Flow past a transversely rotating sphere

4.1.1. Vortex shedding regime

The ‘vortex shedding’ regime has been reported by Giacobello et al. (Reference Giacobello, Ooi and Balachandar2009) and Kim (Reference Kim2009) for the flow past a transversely rotating sphere at $\mathit{Re}\leqslant 300$ and ${\it\Omega}^{\ast }\leqslant 0.30$ . Figure 9 presents the temporal evolution of the vortex shedding flow structures over a shedding period at $\mathit{Re}=500$ and ${\it\Omega}^{\ast }=0.20$ . Isosurfaces are identified using the ${\it\lambda}_{2}$ vortex identification method (Jeong & Hussain Reference Jeong and Hussain1995) by plotting an isosurface value ${\it\lambda}_{2}=-0.0008$ . In the ‘vortex shedding’ regime, the flow comprises horseshoe shape vortices (top of the flow structures) shed from the sphere and induced vortices (bottom of the flow structures) which connect the horseshoe shape vortices together. These flow structures are qualitatively similar to a stationary sphere at $\mathit{Re}=300$ (Johnson & Patel Reference Johnson and Patel1999) and a transversely rotating sphere at low $\mathit{Re}$ and ${\it\Omega}^{\ast }$ (Giacobello et al. Reference Giacobello, Ooi and Balachandar2009; Kim Reference Kim2009) except that the flow loses $(y,z)$ -plane symmetry at certain ${\it\Omega}^{\ast }$ . For a clear illustration of the vortex shedding regime, planar-symmetric flow structures at $\mathit{Re}=500$ and ${\it\Omega}^{\ast }=0.20$ are presented.

Figure 9.

The vortex shedding regime ( $\mathit{Re}=500$ and ${\it\Omega}^{\ast }=0.20$ ): temporal evolution of vortical structures (isosurfaces of ${\it\lambda}_{2}=-0.0008$ ) over a shedding period. Snapshot of out-of-plane vorticities, ${\it\omega}_{i}$ , where the subscript $i=x$ , $y$ and $z$ , in the $(y,z)$ -plane, the $(x,z)$ -plane and a cross-stream plane at $z^{\ast }=5$ ; the projected streamline patterns in the $(y,z)$ -plane are also shown. The projected streamlines are coloured by the out-of-plane velocity, $u_{x}$ .

To further analyse the effect of the sphere rotation on the near-wake flow field, instantaneous out-of-plane vorticities, ${\it\omega}_{i}$ , at five-eighths of the shedding period are presented in the $(x,z)$ -, $(y,z)$ - and cross-stream planes at $z^{\ast }=5.0$ (see figure 9). At this time instant, the shed vortex begins to separate from the sphere. The $x$ -vorticity, ${\it\omega}_{x}$ , in the $(y,z)$ -plane shows long finger-like negative contours on the advancing side of the sphere $(y^{\ast }<0)$ . A small patch of positive ${\it\omega}_{x}$ is also found in the near wake towards the advancing side of the sphere. This is because the sphere rotation expands the positive ${\it\omega}_{x}$ towards $y^{\ast }<0$ (Giacobello et al. Reference Giacobello, Ooi and Balachandar2009). In the wake, a patch of vertical positive ${\it\omega}_{x}$ discharged previously is shown. In the $(x,z)$ -plane, the $y$ -vorticity changes rapidly from positive near the centreline to negative as $x^{\ast }$ increases. The tail of the negative ${\it\omega}_{y}$ rolls up where the shed vortex is located (see the isosurfaces of ${\it\lambda}_{2}$ ). A separate patch of ${\it\omega}_{y}$ is shown further downstream. The location of this patch coincides with the legs of the induced vortex. The location of this patch of ${\it\omega}_{y}$ also indicates that the induced vortex is likely to be formed from the positive $x$ -vorticity in the $(y,z)$ -plane. In the cross-stream plane at $z^{\ast }=5.0$ , the negative $z$ -vorticity $({\it\omega}_{z})$ shows a kidney shape as the plane cuts through the horseshoe vortex (Giacobello et al. Reference Giacobello, Ooi and Balachandar2009).

One of the main features of the ‘vortex shedding’ flow regime is the ‘buildup and release’ of the recirculation bubbles in the near wake (Johnson & Patel Reference Johnson and Patel1999). However, both Johnson & Patel (Reference Johnson and Patel1999) and Giacobello et al. (Reference Giacobello, Ooi and Balachandar2009) mentioned that the presence of the recirculation bubbles can be masked by the vorticity generated by the cylindrical shear layer in the near wake. As a result, to confirm the existence of the recirculation bubbles, the streamline pattern, which is coloured by the out-of-plane velocity $(u_{z})$ , projected in the $(y,z)$ -plane is necessary. The recirculation bubbles are clearly evident in the streamline pattern. As a result of the transverse sphere rotation, the recirculation bubbles are also displaced towards the advancing side of the sphere. The planar-symmetric flow structures result in $u_{x}\approx 0$ .

The effect of the ‘buildup and release’ of the recirculation bubbles in the ‘vortex shedding’ regime on the near-wake flow fields is shown by displaying the near-wake streamline pattern and RMS pressure in the $(y,z)$ -plane in figure 10(a), and the normalised cross-correlations at $(z^{\ast },r^{\ast })=(1.0,0.6)$ in the azimuthal direction in figure 10(b). The cross-correlation function is given as

(4.1) $$\begin{eqnarray}C_{IJ}=\lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}u_{z},\text{}_{I}(t)u_{z},\text{}_{J}(t+{\it\tau})\text{d}t,\end{eqnarray}$$

where the subscripts $I$ and $J$ denote the signal locations of the streamwise velocity with $I,\,J=1$ , 2, 3 and 4, as shown in the inset of figure 10(b). In the present study, the normalised cross-correlation coefficients $R_{IJ}$ are presented, where

(4.2) $$\begin{eqnarray}R_{IJ}\left({\it\tau}\right)=\frac{C_{IJ}\left({\it\tau}\right)}{\sqrt{C_{II}\left(0\right)\,C_{JJ}\left(0\right)}},\end{eqnarray}$$

such that $R_{11}$ stands for the auto-correlation, and $R_{12}$ is the cross-correlation of $u_{z}$ between locations 1 and 2, etc.

Figure 10.

(a) The effect of ‘vortex shedding’ on the near-wake streamline pattern and RMS pressure; (b) normalised cross-correlations, $R$ , versus time delay, ${\it\tau}$ , (see (4.2)) at $z^{\ast }=1.0$ (indicated by the red dashed line in a) for four azimuthal locations (see inset).

Due to the ‘buildup and release’ of the recirculation bubbles, the near wake $(z^{\ast }\approx 0.5{-}2.0)$ is highly unstable. This unstable near wake is clearly shown in the RMS pressure contour plots in the $(y,z)$ -plane where the patch of high RMS pressure coincides with the location of the recirculation bubbles at $z^{\ast }\approx 0.5{-}2.0$ . It should be noted that the RMS pressure is stronger on the advancing side of the sphere $(y^{\ast }<0)$ as the recirculation bubbles are displaced towards the advancing side of the sphere by the transverse sphere rotation. The cross-correlations of $u_{z}$ show that $R_{13}$ is anti-correlated (as $R_{13}({\it\tau}=0)\approx -1$ ) and is ${\rm\pi}$ out-of-phase with $R_{11}$ . This anti-correlated and ${\rm\pi}$ out-of-phase behaviour is attributed to the alternating shed and induced vortices aligned with the $y$ -axis as a result of the ‘buildup and release’ mechanism. On the other hand, $R_{12}$ and $R_{14}$ are less correlated because the recirculation bubbles are ‘buildup and released’ in the $y$ direction.

4.1.2. Shear layer instability regime

The recirculation region on the retreating side of the sphere $(y^{\ast }>0)$ is progressively suppressed as ${\it\Omega}^{\ast }$ increases and the flow structures transform into the ‘shear layer instability’ regime at $\mathit{Re}=500$ and ${\it\Omega}^{\ast }=0.80$ , similarly to those at lower $\mathit{Re}$ (Giacobello et al. Reference Giacobello, Ooi and Balachandar2009; Kim Reference Kim2009). Temporal evolution of the ‘shear layer instability’ vortical structures is shown in figure 11. Differently from the ‘vortex shedding’ regime, the flow structures show double vortical threads in the near wake. These threads tilt and connect to form a one-sided hairpin structure further downstream. Giacobello et al. (Reference Giacobello, Ooi and Balachandar2009) reasoned that the formation of the one-sided vortices was due to the oblique Kelvin–Helmholtz instability.

Figure 11.

Shear layer instability ( $\mathit{Re}=500$ and ${\it\Omega}^{\ast }=0.80$ ): temporal evolution of vortical structures (isosurfaces of ${\it\lambda}_{2}=-0.0008$ ) over a shedding period. Snapshot of out-of-plane vorticities, ${\it\omega}_{i}$ , where the subscript $i=x$ , $y$ and $z$ , in the $(y,z)$ -plane,  $(x,z)$ -plane and a cross-stream plane at $z^{\ast }=5$ ; projected streamline patterns in the $(y,z)$ -plane are also shown. The projected streamlines are coloured by the out-of-plane velocity, $u_{x}$ .

A snapshot of the out-of-plane vorticities, ${\it\omega}_{i}$ , is also presented in figure 11. As ${\it\Omega}^{\ast }$ increases from 0.20 to 0.80, the positive $x$ -vorticity, ${\it\omega}_{x}$ , in the $(y,z)$ -plane is directed towards the advancing side of the sphere and is progressively shorter so that it almost becomes invisible. In the meantime, the increase in rotation rate increases flow shear on the advancing side of the sphere and results in a thinner but stronger negative ${\it\omega}_{x}$ . This negative ${\it\omega}_{x}$ clearly indicates the presence of the shear layer instabilities further downstream and forms isolated patches of negative vortices that represent the one-sided vortices in the flow structures. The shear layer instability is also quantified by Poon et al. (Reference Poon, Ooi, Giacobello and Cohen2013), who compared the Strouhal number of the unsteady shear layer to the analytical Strouhal number from the linear stability analysis. The $y$ -vorticity in the $(x,z)$ -plane shows elongated patches of positive ${\it\omega}_{y}$ near the centreline, but is negative in value near the surface of the sphere. In addition, the contour lines show no sign of waviness. This is because the Kelvin–Helmholtz instability is only observed on the advancing side of the sphere. The cross-stream plane at $z^{\ast }=5.0$ shows a relatively small patch of negative ${\it\omega}_{z}$ . However, the ${\it\omega}_{z}$ contours are more concentrated as the Kelvin–Helmholtz instability redistributes the shear layer vorticity (Giacobello et al. Reference Giacobello, Ooi and Balachandar2009).

The instantaneous projected streamline pattern is presented at the bottom of the figure. Within the ‘shear layer instability’ regime, the recirculation bubbles are completely suppressed. The fluid on the retreating side of the sphere is vectored towards the advancing side and creates a shearing motion on the advancing side. The shear layer eventually becomes unstable and releases small vortices, as shown in the $x$ -vorticity in the $(y,z)$ -plane.

The shear layer is clearly evident in figure 12(a), which shows the temporal evolution of the streamline pattern. The shear layer is relatively stable in the near wake and instability only begins at $z^{\ast }\geqslant 2.0$ . As a result, the high RMS pressure is pushed downstream and is only evident near the onset of the shear layer instability (waviness of the streamlines at $z^{\ast }\geqslant 2.0$ ). The transformation of flow from the ‘vortex shedding’ regime to the ‘shear layer instability’ regime also leads to a substantial change in the cross-correlations of $u_{z}$ , as shown in figure 12(b). Within the ‘shear layer instability’ regime, $R_{12}$ and $R_{14}$ become more anti-correlated than $R_{13}$ . In addition, $R_{13}$ is less anti-correlated for the ‘shear layer instability’ regime compared with the ‘vortex shedding’ regime $(R_{13\text{-}vortex\;shedding}>R_{13\text{-}shear\;layer\;instab.}\text{ at }{\it\tau}=0)$ . The physical reason for this change in cross-correlations can be attributed to the one-sided vortices that scale with the shear layer on the advancing side of the sphere (Giacobello et al. Reference Giacobello, Ooi and Balachandar2009). As a result, the flow on the retreating side of the sphere is less affected by the instability of the shear layer. Similarly, the increasing anti-correlated $R_{12}$ and $R_{14}$ can be attributed to the locations of 2 and 4 being closer to the shear layer.

Figure 12.

(a) The effect of ‘shear layer instability’ on the near-wake streamline pattern and RMS pressure; (b) normalised cross-correlations, $R$ , versus time delay, ${\it\tau}$ , (see (4.2)) at $z^{\ast }=1.0$ (indicated by the red dashed line in a) for four azimuthal locations (see inset).

4.1.3. Shear layer–stable foci regime

A newly discovered flow regime is observed above the ‘shear layer instability’ regime at $\mathit{Re}=500$ and ${\it\Omega}^{\ast }=1.00$ . This flow regime is denoted the ‘shear layer–stable foci’ regime. Figure 13 shows the temporal evolution of the flow structures with the instantaneous out-of-plane vortices and projected streamlines for this newly discovered regime. The flow structures become chaotic in the ‘shear layer–stable foci’ regime. Vortex loops roll up in different directions and form a ‘zigzag’ pattern in the wake. Although the vortex loops are no longer rolling up in a unidirectional manner, the orientation of these vortex loops is far from random. Vortex loops are either rolling up towards the advancing side of the sphere or at a slight angle. This vortex loop roll-up at an angle contributes to the sideways oriented vortex loops further downstream.

Figure 13.

‘Shear layer–stable foci’ regime ( $\mathit{Re}=500$ and ${\it\Omega}^{\ast }=1.00$ ): temporal evolution of vortical structures (isosurfaces of ${\it\lambda}_{2}=-0.0008$ ) over a shedding period. Snapshot of out-of-plane vorticities, ${\it\omega}_{i}$ , where the subscript $i=x$ , $y$ and $z$ , in the $(y,z)$ -plane,  $(x,z)$ -plane and a cross-stream plane at $z^{\ast }=5$ ; projected streamline patterns in the $(y,z)$ -plane are also shown. The projected streamlines are coloured by the out-of-plane velocity, $u_{x}$ .

The chaotic pattern of the flow structures can be explained in detail using the out-of-plane vorticities ${\it\omega}_{i}$ . A further increase of ${\it\Omega}^{\ast }$ from 0.80 to 1.00 shows an intensified (highly negative) shear layer vorticity ${\it\omega}_{x}$ on the advancing side of the sphere in the $(y,z)$ -plane. The shear layer vorticity is significantly shortened at this rotation rate and vortices of negative ${\it\omega}_{x}$ are released from the shear layer as early as $z^{\ast }\approx 1.0$ . Furthermore, a patch of positive ${\it\omega}_{x}$ is formed near the centreline at $z^{\ast }\approx 1.0$ below the first vortex that is released from the shear layer instability. This positive ${\it\omega}_{x}$ is detached from the lee side of the sphere and interacts with the negative ${\it\omega}_{x}$ to form a chaotic flow pattern in the wake comprising small patches of both positive and negative ${\it\omega}_{x}$ vortices. The flow unsteadiness near the centreline is also demonstrated by the waviness of positive ${\it\omega}_{y}$ in the $(x,z)$ -plane. As most of the vortices are released from the shear layer on the advancing side of the sphere, this unsteady ${\it\omega}_{y}$ motion near the centreline at $z^{\ast }\approx 2.0$ may be localised. In the cross-stream plane at $z^{\ast }=5.0$ , the interaction of the positive ${\it\omega}_{x}$ with the negative ${\it\omega}_{x}$ from the shear layer has led to small patches of positive and negative ${\it\omega}_{z}$ .

The instantaneous projected streamline pattern is shown at the bottom of the figure. The streamline pattern has a topology similar to those at ${\it\Omega}^{\ast }=0.80$ except that a stable focus (Chong, Perry & Cantwell Reference Chong, Perry and Cantwell1990) is formed on the advancing side of the sphere at $z^{\ast }\approx 1.0$ . The formation of this stable focus is different from the recirculation bubbles shown in the ‘vortex shedding’ regime (refer to figure 9). One of the major differences is that this stable focus is located near the onset of the shear layer instability rather than on the lee side of the sphere. The presence of the stable focus near the onset of the shear layer instability leads to an increase in flow unsteadiness in the near wake. The increase in flow unsteadiness below the stable focus at $z^{\ast }\approx 1.0$ explains the reappearance of the positive ${\it\omega}_{x}$ in the $(y,z)$ -plane. As a result of this stable focus near the onset of shear layer instability, small patches of positive and negative ${\it\omega}_{x}$ vortices in the $(y,z)$ -plane are thus shown in the wake. The presence of the stable focus also contributes to the ‘zigzag’ pattern of the vortex loops in the wake as there is a significant increase in velocity $(u_{x})$ both into and out of the $(y,z)$ -plane.

Figure 14(a) shows the temporal evolution of the stable focus over a shedding period. The stable focus oscillates in size as it convects downstream and results in highly unsteady flow on the advancing side of the sphere at $z^{\ast }>0.5$ . The increase in flow unsteadiness in this region leads to a substantial increase in pressure fluctuations on the advancing side of the sphere at $z^{\ast }>0.5$ , as shown in the RMS pressure contours. The cross-correlations $R_{12}$ , $R_{13}$ and $R_{14}$ in figure 14(b) show that the flow no longer comprises a single frequency in this flow regime. The second frequency may be linked to the existence of the stable focus. Apart from showing higher frequency oscillations, the cross-correlations for the ‘shear layer–stable foci’ regime show similar behaviour to the ‘shear layer instability’ regime, i.e.  $R_{12}$ and $R_{14}$ are more anti-correlated than $R_{13}$ as a result of the shear layer instability on the advancing side of the sphere. In conclusion, it is observed that $R_{12}$ and $R_{14}$ become more anti-correlated than $R_{13}$ as the flow transforms from the ‘vortex shedding’ regime to either the ‘shear layer instability’ or the ‘shear layer–stable foci’ regime.

Figure 14.

(a) The effect of ‘shear layer–stable foci’ on the near-wake streamline pattern and RMS pressure; (b) normalised cross-correlations, $R$ , versus time delay, ${\it\tau}$ , (see (4.2)) at $z^{\ast }=1.0$ (indicated by the red dashed line in a) for four azimuthal locations (see inset).

4.2. Effect of Reynolds number on the shear layer–stable foci regime

Figure 15 presents the flow past a transversely rotating sphere at $\mathit{Re}=1000$ and ${\it\Omega}^{\ast }=1.20$ . As the vortices are scaled with the shear layer in the ‘shear layer–stable foci’ regime, increasing $\mathit{Re}$ and ${\it\Omega}^{\ast }$ result in smaller vortices in the wake. On the other hand, the flow structures display some kind of large-scale shedding pattern qualitatively similar to that in the ‘vortex shedding’ regime. Although the large-scale shedding pattern of the ‘roll-up and release’ of the small vortices from the Kelvin–Helmholtz instability was also reported for a stationary sphere at $\mathit{Re}=3700$ (Yun, Kim & Choi Reference Yun, Kim and Choi2006), the large-scale shedding pattern observed here is attributed to the presence of the stable focus near the onset of shear layer instability and will be discussed in detail.

Figure 15.

Effect of Reynolds number on the ‘shear layer–stable foci’ ( $\mathit{Re}=1000$ and ${\it\Omega}^{\ast }=1.20$ ) vortical structures (isosurfaces of ${\it\lambda}_{2}=-0.0008$ ) over a shedding period. Snapshot of out-of-plane vorticities, ${\it\omega}_{i}$ , where the subscript $i=x$ , $y$ and $z$ , in the $(y,z)$ -plane,  $(x,z)$ -plane and a cross-stream plane at $z^{\ast }=5$ ; projected streamline patterns in the $(y,z)$ -plane are also shown. The projected streamlines are coloured by the out-of-plane velocity, $u_{x}$ .

Before we present the instantaneous streamline pattern and discuss the effect of $\mathit{Re}$ on the stable focus, the out-of-plane vorticities $({\it\omega}_{i})$ are described. In the $(y,z)$ -plane, the negative $x$ -vorticity shear layer is substantially shortened. The patch of positive ${\it\omega}_{x}$ is also brought upstream and is nearly attached to the downstream pole of the sphere. This $x$ -vorticity patch is drawn upward in the negative $y$ direction and interacts with the negative vorticity near the onset of the shear layer instability. The interaction between the positive and negative vorticity patches draws the $x$ -vorticity downward. Further downstream, other patches of vorticity are shown vectoring upward again forming a vortex-shedding-like pattern in the $(y,z)$ -plane. In the $(x,z)$ -plane, the small patch of $y$ -vorticity $({\it\omega}_{y})$ in the near wake clearly indicates the shortened shear layer at this $\mathit{Re}$ and ${\it\Omega}^{\ast }$ . A patch of positive ${\it\omega}_{y}$ begins to roll up in the horizontal direction in the wake $(z^{\ast }\approx 2.0{-}3.0)$ . In the cross-stream plane at $z^{\ast }=5.0$ , only a small patch of the negative ${\it\omega}_{y}$ is observed.

The instantaneous projected streamline pattern is also shown in the figure. The streamlines on the retreating side of the sphere $(y^{\ast }>0)$ are clearly vectored towards the advancing side of the sphere, suppressing any shear layer instability of the Kelvin–Helmholtz type on the retreating side, as observed for flow past a stationary sphere at $\mathit{Re}=1000$ (Kim & Durbin Reference Kim and Durbin1988; Sakamoto & Haniu Reference Sakamoto and Haniu1990; Bonneton & Chomax Reference Bonneton and Chomax1992; Tomboulides & Orszag Reference Tomboulides and Orszag2000). These streamlines join those of the advancing side and result in shear layer instability of the Kelvin–Helmholtz type. Due to the high local Reynolds number $(\mathit{Re}_{l}=2200)$ , the onset of shear layer instability is pushed further upstream and the stable focus (Chong et al. Reference Chong, Perry and Cantwell1990) is formed immediately behind the sphere. In addition, it is shown in the figure that the previous stable focus diffuses quickly as it convects downstream, drawing the streamlines towards the bottom of the plane. This explains the downward motion of the $x$ -vorticity at $z^{\ast }\approx 1.0{-}2.0$ . However, the surrounding flow is continuously drawn upward by the sphere rotation; the flow structures therefore exhibit a large-scale shedding pattern.

Figure 16.

(a) Phase diagram $(C_{Ly},C_{Lx})$ and (b) phase diagram $(C_{L},C_{D})$ for a transversely rotating sphere at $\mathit{Re}=500$ .

Figure 17.

(a) Phase diagram $(C_{Ly},C_{Lx})$ and (b) phase diagram $(C_{L},C_{D})$ for a transversely rotating sphere at $\mathit{Re}=1000$ .

4.3. Hydrodynamic force coefficients for a transversely rotating sphere at $\mathit{Re}=500{-}1000$

The dynamic behaviour of the lift and drag coefficients acting on a transversely rotating sphere at $\mathit{Re}=500$ and 1000 is reported using phase diagrams of $(C_{Ly},C_{Lx})$ and $(C_{L},C_{D})$ in figures 16 and 17. All flow solutions are unsteady at these Reynolds numbers, but some solutions may appear to be steady due to the scaling of the phase diagram. These solutions are represented by a solid delta (▴). At $\mathit{Re}=500$ and ${\it\Omega}^{\ast }=0$ , $C_{Lx}$ and $C_{Ly}$ change irregularly and show a preferential orientation towards the positive $C_{Lx}$ -axis within a finite period of time; this was also reported by Mittal, Wilson & Najjar (Reference Mittal, Wilson and Najjar2002). As the sphere rotates, the preferential direction of lift changes from the positive $C_{Lx}$ -axis to the positive $C_{Ly}$ -axis due to the pressure differential created by a transversely rotating sphere. Increasing ${\it\Omega}^{\ast }$ decreases the oscillation magnitudes of $C_{Ly}$ for ${\it\Omega}^{\ast }\leqslant 0.60$ . As ${\it\Omega}^{\ast }$ increases from 0.50 to 0.60, the recirculation bubble on the retreating side of the sphere is almost suppressed. Therefore, although the flow structures are qualitatively similar as they are both within the ‘vortex shedding’ regime, the smaller recirculation bubble induces lower pressure fluctuation in the near wake and leads to a decrease in $C_{Ly}$ oscillation magnitudes. This decrease in lift coefficient fluctuation is most obvious as the flow enters the ‘shear layer instability’ regime where the recirculation bubbles are completely suppressed (see figure 12). Further increase in ${\it\Omega}^{\ast }$ to 1.00 leads to the formation of stable foci. These stable foci oscillate vigorously as they travel downstream (as shown in figure 14) and increase the pressure fluctuation in the near wake. As a result, both $C_{Lx}$ and $C_{Ly}$ oscillation magnitudes increase substantially.

Similar behaviour is observed for $C_{L}$ and $C_{D}$ at $\mathit{Re}=500$ . At ${\it\Omega}^{\ast }\leqslant 0.60$ (within the ‘vortex shedding’ regime), both $C_{L}$ and $C_{D}$ show large oscillations. Here, $C_{L}$ and $C_{D}$ decrease their oscillation magnitudes as the flow enters the ‘shear layer instability’ regime at ${\it\Omega}^{\ast }=0.80$ , but increase their oscillation magnitudes dramatically at ${\it\Omega}^{\ast }=1.00$ due to the oscillations of the stable foci.

At $\mathit{Re}=1000$ and ${\it\Omega}^{\ast }=0.00$ , the lift coefficients $C_{Ly}$ and $C_{Lx}$ again change irregularly with time but do not show a preferential direction. For a transversely rotating sphere at ${\it\Omega}^{\ast }\leqslant 0.60$ , even though the flow is within the ‘vortex shedding’ regime, the oscillation amplitudes of $C_{Ly}$ and $C_{Lx}$ are relatively small. Above ${\it\Omega}^{\ast }=0.80$ , the flow transforms to the ‘shear layer–stable foci’ regime and there is a slight increase in both $C_{Lx}$ and $C_{Ly}$ oscillation amplitudes. Their oscillation amplitudes increase dramatically for ${\it\Omega}^{\ast }\geqslant 1.00$ . The reason for the smaller oscillation amplitudes of $C_{Ly}$ at ${\it\Omega}^{\ast }\leqslant 0.60$ is attributed to the presence of the shear layer instability on the advancing side of the sphere. Giacobello et al. (Reference Giacobello, Ooi and Balachandar2009), Kim (Reference Kim2009) and Poon et al. (Reference Poon, Ooi, Giacobello and Cohen2013) showed that eddies released from the Kelvin–Helmholtz instability of the shear layer reduce the unsteadiness of the near-wake flow field. On the other hand, the oscillations of the stable foci contribute to the increase in $C_{Lx}$ and $C_{Ly}$ oscillation amplitudes. As ${\it\Omega}^{\ast }$ increases, these stable foci oscillate more vigorously and thus large oscillation amplitudes of $C_{Lx}$ and $C_{Ly}$ are observed.

Figure 18.

Time-averaged force coefficients for the flow past a stationary and transversely rotating sphere at $\mathit{Re}=500{-}1000$ : (a) $\overline{C_{Lx}}$ against ${\it\Omega}^{\ast }$ ; (b) $\overline{C_{Ly}}$ against ${\it\Omega}^{\ast }$ ; (c) $\overline{C_{D}}$ against ${\it\Omega}^{\ast }$ . The effect of integration time on the time-averaged force coefficients is shown in the insets for a sample parameter $(\mathit{Re}=500\text{ and }{\it\Omega}^{\ast }=0.60)$ . Time-averaged force coefficients are displayed at half integration time and full integration time.

Both $C_{L}$ and $C_{D}$ at $\mathit{Re}=1000$ show low oscillation amplitudes within the ‘vortex shedding’ regime $({\it\Omega}^{\ast }\leqslant 0.60)$ . This is again related to the formation of the shear layer instability of the Kelvin–Helmholtz type on the advancing side of the sphere. On the other hand, for ${\it\Omega}^{\ast }\geqslant 0.80$ the stable foci oscillate and result in a substantial increase in the oscillation amplitudes of both $C_{L}$ and $C_{D}$ .

4.4. Time-averaged force coefficients for a transversely rotating sphere at $\mathit{Re}=500{-}1000$

The time-averaged force coefficients acting on a stationary and transversely rotating sphere at $\mathit{Re}=500$ and 1000 are presented in figure 18. In all cases, the force coefficients are averaged over $t^{\ast }=150$ , which is equivalent to approximately 25 shedding periods for the flow past a stationary sphere at $\mathit{Re}=500{-}1000$ . The effect of time integration is demonstrated in the insets of figure 18 for a sample parameter at $\mathit{Re}=500$ and ${\it\Omega}^{\ast }=0.60$ . Convergence criteria are set such that the variation in time-averaged force coefficients is within 3 % of the value obtained at half the integration time interval. The time-averaged lift coefficient in the $x$ direction, $\overline{C_{Lx}}$ , is presented in figure 18(a); $\overline{C_{Lx}}\approx 0$ for all $\mathit{Re}$ and ${\it\Omega}^{\ast }$ considered beside a stationary sphere at $\mathit{Re}=500$ and a transversely rotating sphere at $\mathit{Re}=1000$ and ${\it\Omega}^{\ast }=1.20$ . The non-zero $\overline{C_{Lx}}$ for a stationary sphere at $\mathit{Re}=500$ may be attributed to flow structures showing a preferential orientation. Similar findings were also reported by Bagchi, Ha & Balachandar (Reference Bagchi, Ha and Balachandar2001) and Mittal et al. (Reference Mittal, Wilson and Najjar2002). Such preferential orientation is also observed for $\mathit{Re}=1000$ and ${\it\Omega}^{\ast }=1.20$ as a result of the vigorous oscillations of the stable foci near the onset of shear layer instability and leads to a non-zero $\overline{C_{Lx}}$ . However, it is important to note that this preferential orientation of the flow may not exist over a longer integration time and thus results in $\overline{C_{Lx}}\rightarrow 0$ (Yun et al. Reference Yun, Kim and Choi2006). Although $\overline{C_{Lx}}$ may approach zero as the preferential orientation ceases over a longer integration time, a transversely rotating sphere always induces flow in the $y$ directions due to the ‘Robins–Magnus’ effect (see figures 9, 11, 13 and 15). As a result, time integration over 25 shedding periods that results in 3 % variation in $\overline{C_{Ly}}$ is considered adequate. In general, $\overline{C_{Ly}}$ increases with ${\it\Omega}^{\ast }$ , but decreases with $\mathit{Re}$ . However, the sensitivity of $\overline{C_{Ly}}$ to ${\it\Omega}^{\ast }$ decreases as ${\it\Omega}^{\ast }$ increases. Poon et al. (Reference Poon, Ooi, Giacobello and Cohen2010) reasoned that the decrease in sensitivity is due to the physical limit on the flow separation. Experimental results from Tsuji et al. (Reference Tsuji, Morikawa and Mizuno1985) as well as the correlation models of Tanaka et al. (Reference Tanaka, Yamagata and Tsuji1990) and Loth (Reference Loth2008) are also included. It is observed that experimental results reported a smaller $\overline{C_{Ly}}$ for a given ${\it\Omega}^{\ast }$ . On the other hand, Tanaka et al.’s correlation model was derived for subcritical $\mathit{Re}$ , but was shown to show good agreement with $\overline{C_{Ly}}$ at low $\mathit{Re}$ by Loth (Reference Loth2008). It is also important to mention that Loth’s model, although it gives a smaller $\overline{C_{Ly}}$ compared with our numerical results, provides a good agreement with the experimental results as it is mainly derived from the experimental data. The time-averaged resultant drag coefficient, $\overline{C_{D}}$ , is presented in figure 18(c). In brief, $\overline{C_{D}}$ increases with ${\it\Omega}^{\ast }$ but decreases with $\mathit{Re}$ .

The smaller experimental $\overline{C_{Ly}}$ may be attributed to the difficulties in maintaining a constant sphere rotation rate as the rotating sphere travels through a stationary viscous medium (Tsuji et al. Reference Tsuji, Morikawa and Mizuno1985; Tri et al. Reference Tri, Oesterle and Deneu1990). Tsuji et al. (Reference Tsuji, Morikawa and Mizuno1985) estimated that the sphere rotation decreases at a rate proportional to the viscous torque, $T$ , on the moment of inertia, $I$ , of the sphere, i.e. $\text{d}{\it\Omega}/\text{d}t=-T/I$ . However, the viscous torque acting on the sphere was only estimated based on the numerical results of Dennis, Singh & Ingham (Reference Dennis, Singh and Ingham1980). Bagchi & Balachandar (Reference Bagchi and Balachandar2002) performed a similar study numerically on the motion of a sphere in a linear shear flow and concluded that the sphere shows a faster decay rate for its rotating motion than its translation motion at finite Reynolds numbers. The ratio of the rotation decay, ${\it\tau}_{r}$ , to the translation decay, ${\it\tau}_{t}$ , lies between 0 and 0.3. Although the above expression was derived for a sphere that is free to translate and rotate in a linear shear flow, one would also expect similar decay behaviour for a sphere translating and rotating in a stationary viscous medium. Other uncertainties in measuring the sphere rotation rate, the sphere trajectory and its axis of rotation may contribute to the difference between experimental and numerical lift coefficients. In fact, Oesterlé & Dinh (Reference Oesterlé and Dinh1998) stated an error of 20 % in determining the sphere rotation rate. Tsuji et al. (Reference Tsuji, Morikawa and Mizuno1985) also reported a wide range of sphere rotation rates from their experimental set-up and the reported rotation rates are averaged over 100 trials. Furthermore, Bagchi & Balachandar (Reference Bagchi and Balachandar2003) showed that the effect of free stream disturbance, which is often difficult to remove experimentally, displays a small effect on the lift coefficient for a non-rotating sphere. The effect of free stream disturbance may become more pronounced as the sphere rotates due to the wake structures being sensitive to the modest changes in $\mathit{Re}$ and ${\it\Omega}^{\ast }$ .

The time-averaged resultant force coefficients $(\overline{C_{D}}\text{ and }\overline{C_{Ly}})$ versus Reynolds numbers for the flow past a stationary and transversely rotating sphere are presented in figure 19. The numerical data for laminar Reynolds numbers obtained in Giacobello et al. (Reference Giacobello, Ooi and Balachandar2009), Kim (Reference Kim2009) and Poon et al. (Reference Poon, Ooi, Giacobello and Cohen2010), experimental data from Tsuji et al. (Reference Tsuji, Morikawa and Mizuno1985), Tanaka et al. (Reference Tanaka, Yamagata and Tsuji1990) and Tri et al. (Reference Tri, Oesterle and Deneu1990), and the lift correlation model by Loth (Reference Loth2008) are also superimposed onto the figure. In general, the resultant drag coefficient, $\overline{C_{D}}$ , increases with ${\it\Omega}^{\ast }$ but decreases with $\mathit{Re}$ . On the other hand, although the time-averaged lift coefficient, $\overline{C_{Ly}}$ , increases with the dimensionless rotation rate, ${\it\Omega}^{\ast }$ , it shows two different behaviours (highlighted in figure 19 a by white and grey colour) with increasing $\mathit{Re}$ . At low ${\it\Omega}^{\ast }\ ({\it\Omega}^{\ast }\leqslant 0.30)$ , $\overline{C_{Ly}}$ increases initially and decreases after reaching a maximum as $\mathit{Re}$ increases. However, for higher rotation rate $({\it\Omega}^{\ast }\geqslant 0.50)$ , all calculations show a monotonic decrease in $\overline{C_{Ly}}$ with $\mathit{Re}$ . It is also important to remember that the flow structures for a stationary sphere may show a preferential direction over the time integration of 25 shedding periods. As a result of this historical bias in data collected, $\overline{C_{Ly}}$ shows a significant magnitude at these considered Reynolds numbers.

Figure 19.

Time-averaged resultant force coefficients against Reynolds number for the flow past a stationary and transversely rotating sphere: (a) $\overline{C_{D}}$ and (b) $\overline{C_{Ly}}$ . The grey box highlights the change in behaviour in $\overline{C_{Ly}}$ with $\mathit{Re}$ .

Figure 20.

RMS force coefficients for the flow past a stationary and transversely rotating sphere at $\mathit{Re}=500{-}1000$ : (a) ${C_{Lx}}^{\prime }$ against ${\it\Omega}^{\ast }$ ; (b) ${C_{Ly}}^{\prime }$ against ${\it\Omega}^{\ast }$ ; (c) ${C_{D}}^{\prime }$ against ${\it\Omega}^{\ast }$ .

Figure 21.

Strouhal number, $St$ , versus dimensionless rotation rate, ${\it\Omega}^{\ast }$ , for a transversely rotating sphere.

In early sections of this paper, it was mentioned that the stable foci oscillate vigorously and lead to a significant increase in the near-wake RMS pressure fluctuations and thus higher oscillation magnitudes for the force coefficients. Here, we quantify this effect by presenting the RMS force coefficients. In figure 20(a), the RMS force coefficient fluctuation, ${C_{Lx}}^{\prime }$ , in the $x$ direction is presented. Here, ${C_{Lx}}^{\prime }>0$ for all cases considered except for $\mathit{Re}=500$ and ${\it\Omega}^{\ast }=0.20$ . It should be noted that for $\mathit{Re}=500$ and ${\it\Omega}^{\ast }=0.80$ , ${C_{Lx}}^{\prime }$ is slightly above zero. In addition, it is observed that as the flow enters the newly discovered ‘shear layer–stable foci’ regime, there is a substantial increase in ${C_{Lx}}^{\prime }$ for both Reynolds numbers. Here, ${C_{Lx}}^{\prime }=0$ at $\mathit{Re}=500$ and ${\it\Omega}^{\ast }=0.20$ due to the fact that the flow remains planar-symmetric about the $(y,z)$ -plane. For ${C_{Ly}}^{\prime }$ at $\mathit{Re}=500$ , it is evident in figure 20(b) that with increasing ${\it\Omega}^{\ast }$ , ${C_{Ly}}^{\prime }$ decreases and reaches a minimum at ${\it\Omega}^{\ast }=0.80$ ; ${C_{Ly}}^{\prime }$ increases dramatically at ${\it\Omega}^{\ast }=1.00$ . For $\mathit{Re}=1000$ , ${C_{Ly}}^{\prime }$ also decreases initially with ${\it\Omega}^{\ast }$ . However, ${C_{Ly}}^{\prime }$ increases substantially at ${\it\Omega}^{\ast }\geqslant 0.80$ when the flow enters the ‘shear layer–stable foci’ regime. The results for $\mathit{Re}=640$ and 800 show a similar increase in ${C_{Ly}}^{\prime }$ for ${\it\Omega}^{\ast }\geqslant 0.80$ . A similar behaviour is also observed for ${C_{D}}^{\prime }$ in figure 20(c). Hence, it can be concluded that the presence of stable foci leads to an increase in the RMS force coefficients for the Reynolds numbers considered.

The Strouhal number, $St$ , versus the dimensionless rotation rate, ${\it\Omega}^{\ast }$ , is presented in figure 21. The Strouhal number is calculated by averaging the frequency of the streamwise velocity, $u_{z}$ , at $(z^{\ast },r^{\ast })=(2.5,0.6)$ in the azimuthal direction. Only the dominant frequency, which is the frequency corresponding to the most energetic mode, is presented. For $\mathit{Re}=500$ , the Strouhal number monotonically increases for ${\it\Omega}^{\ast }\leqslant 0.80$ before decreasing slightly. There is also a substantial increase in $St$ for ${\it\Omega}^{\ast }=0.80$ compared with lower ${\it\Omega}^{\ast }$ . For $\mathit{Re}=1000$ , $St$ increases for ${\it\Omega}^{\ast }\leqslant 0.80$ . For ${\it\Omega}^{\ast }=1.00$ , there is a sudden decrease in $St$ , but it increases again for ${\it\Omega}^{\ast }=1.20$ . For $\mathit{Re}=640$ and 800, high $St$ is observed at ${\it\Omega}^{\ast }=0.60$ . Here, $St$ decreases significantly at $\mathit{Re}=640$ and ${\it\Omega}^{\ast }=0.80$ . At ${\it\Omega}^{\ast }=1.00$ , both values of $\mathit{Re}$ show similar values of $St$ and are closer to the one at $\mathit{Re}=1000$ .