3.1 Qualitative description of flow regimes

At $Re=250$, the flow behind the streamwise rotating sphere is unsteady for all values of $\unicode[STIX]{x1D6FA}>0$ investigated, and several wake regimes of increasing complexity are found as the rotation parameter increases, as shown in figure 2, where instantaneous plots of vortical structures are depicted for selected values of $\unicode[STIX]{x1D6FA}$ by means of isosurfaces of the $Q$-quantity (corresponding movies are available in the supplemental material, available at https://doi.org/10.1017/jfm.2020.342).

The dynamics associated to each wake regime is studied with the help of figure 3, where time histories of the drag coefficient, $c_{x}(t)$, lift coefficient in the $y$-direction, $c_{y}(y)$, and phase diagrams ($c_{z},c_{y}$) and ($c_{l},c_{x}$) are included (note that the shown range $t\geqslant 0$ corresponds to the initial, statistically converged part of each simulation). The wake topology is further illustrated in figure 4 by means of contours of standard fluctuations of the pressure coefficient, $C_{p}^{\prime }$, and time-averaged velocity streamlines at $y=0$.

The non-rotating sphere ($\unicode[STIX]{x1D6FA}=0$) displays a steady bifid wake with two threads of opposite axial vorticity, $\unicode[STIX]{x1D71B}_{x}$, which are symmetric with respect to the $y=0$ plane (note that the side view in figure 2 presents a conical perspective). As displayed in figure 3, such a steady state regime is characterized by constant values of drag and lift coefficients (note that $c_{y}=0$ due to the horizontal reflectional symmetry), and, therefore, phase diagrams $(c_{y},c_{z})$ and $(c_{l},c_{x})$ correspond to single points. A positive total lift is induced by the flow asymmetry on the vertical plane $z=0$ (see figure 4a).

Figure 3.

Force coefficients for a streamwise rotating sphere at $Re=250$. (a) Time histories of the drag coefficient, $c_{x}(t)$ (solid line), and lift coefficient in the $y$-direction, $c_{y}(t)$ (dashed line). (b) Phase diagrams of transversal forces ($c_{z},c_{y}$). (c) Phase diagrams of the total lift versus drag ($c_{l},c_{x}$). Insets in (b) display a zoomed part of the phase diagrams.

When the streamwise rotation is applied, the wake becomes unsteady and the reflectional symmetry is broken. For instance, at $\unicode[STIX]{x1D6FA}=0.2$ (figure 2) the sphere rotation promotes a stronger positive vorticity thread, which displays a spiral pattern rolling in the opposite sense to that of the body swirl (this suggests that the flow dynamics may stem from the destabilization of an helical mode with azimuthal wavenumber $m=-1$, as highlighted by Jiménez-González, Manglano-Villamarín & Coenen Reference Jiménez-González, Manglano-Villamarín and Coenen2019), while a shroud of negative axial vorticity develops around the surface. This vortical structure is found to rotate, without any variation of shape, with a constant angular velocity whose value is smaller than the sphere’s rotation rate. This periodic or frozen regime, which has been previously identified for different rotating bodies (Kim & Choi Reference Kim and Choi2002; Jiménez-González et al. Reference Jiménez-González, Sanmiguel-Rojas, Sevilla and Martínez-Bazán2013; Pier Reference Pier2013), produces a constant value of the drag coefficient, $c_{x}(t)$, but periodic time histories of transverse coefficients, e.g. $c_{y}(t)$. This periodic rotation is identified in the phase diagram of lift components ($c_{y},c_{z}$) in figure 3(b), as a closed perfect circle centred at the origin (see $\unicode[STIX]{x1D6FA}=0.4$). Its radius indicates a constant value of $c_{l}$, which translates into a single point in the phase diagram ($c_{l},c_{x}$), as shown in figure 3(c), where it can be also observed that an increase of the value of $\unicode[STIX]{x1D6FA}$ leads to a growth of $c_{x}$. As displayed in figure 4(b), the stronger pressure fluctuations are mainly concentrated around the sphere and at the shear layer close to the downstream end of the recirculating bubble, due to the rotation of the asymmetric shroud and wavy vortex thread.

Figure 4.

Contours of standard fluctuations of the pressure coefficient, $C_{p}^{\prime }$, along with time-averaged velocity streamlines at $y=0$ for selected values of the rotation parameter $\unicode[STIX]{x1D6FA}$ at $Re=250$.

This scenario is preserved approximately until $\unicode[STIX]{x1D6FA}=0.6$, where a new regime, characterized by the concomitant rotation of spiral structures and vortex shedding, appears, as depicted in figure 2 for $\unicode[STIX]{x1D6FA}=1.0$. In particular, the increase of $\unicode[STIX]{x1D6FA}$ strengthens the former wavy positive vorticity thread, which adopts a helical structure, being a consequence of the conservation of azimuthal momentum, which requires that fluid particles at larger values of the radial coordinate rotate more slowly. Simultaneously, there is a shedding of vortices from the shroud of negative vorticity, which are encircled by the outer spiral. Such inner vortices are convected downstream and grow into larger structures, intertwining with the helical thread.

This translates into an oscillating time history of drag coefficient $c_{x}(t)$ (see figure 3a), and a shorter oscillation period in the lift component $c_{y}(t)$, due to the stronger rotation rate of the sphere. The phase diagram of lift components describes a loop which is no longer perfectly circular and features several orbits (see figure 3b insets), indicating quasi-periodic flow dynamics, i.e. the vortical structures vary their shape as they spin. Note that the total lift value fluctuates in time with a larger magnitude than the drag, thus leading to a phase diagram ($c_{l},c_{x}$) defined by elongated loops (figure 3c). Moreover, as $\unicode[STIX]{x1D6FA}$ increases, the fluctuations of $c_{l}$ becomes more acute than those from $c_{x}$ (an exception being $\unicode[STIX]{x1D6FA}=0.8$, as it will be later discussed), whose mean value also increases with the rotation rate.

The associated shedding intensifies the pressure fluctuations in the wake, as depicted in figure 4(c), and especially close to the rear stagnation point of the recirculation bubble.

The quasi-periodic regime prevails up to $\unicode[STIX]{x1D6FA}=1.6$, where the wake undergoes a sudden transition to an irregular state, as depicted in figure 2. The spatial structure is more complex, featuring two irregular outer spirals (of positive vorticity $\unicode[STIX]{x1D71B}_{x}$) in the near wake, that become tangled as they move downstream. Additionally, there is a pair of vortices that detach from the shroud of negative $\unicode[STIX]{x1D71B}_{x}$ surrounding the sphere (see also pressure fluctuations close to the separation points in figure 4d) that wrap around the outer spirals while convected downstream, although they are soon dissipated. A similar double-threaded helical structure has been recently observed experimentally by Skarysz et al. (Reference Skarysz, Rokicki, Goujon-Durand and Wesfreid2018) for $\unicode[STIX]{x1D6FA}=1.6$. Now, fluctuating time histories of $c_{x}$ and $c_{y}$ becomes irregular (see figure 3a), leading to complex phase diagrams of ($c_{y},c_{z}$) and ($c_{l},c_{x}$) in figure 3(b,c), anticipating the existence of multiple frequency components. In general, the vortical structures of this regime feature a higher eccentricity than those observed earlier, as depicted in figure 4(d), on account of the larger centrifugal driving force induced by $\unicode[STIX]{x1D6FA}$. As a consequence, the amplitude of lift components and total lift are considerably larger than those reported for the previous regime (see figure 3b,c). In view of such features, we will sometimes refer to such a regime as high-helical irregular, to identify the irregular swirl and shedding of double spiral structures.

As the rotation rate increases above $\unicode[STIX]{x1D6FA}=2$, the wake arranges itself into a more coherent double helical structure in the near wake, as observed in figure 2 for $\unicode[STIX]{x1D6FA}=2.2$. The wake rotation now features a quasi-periodic time history of the lift component $c_{y}$ and a fluctuating drag $c_{x}$ (figure 3a). A closer analysis of $c_{y}$ shows that there are two characteristic oscillating periods, which, as it will be later detailed, are related to the swirl of two different spatial structures in the wake, namely single and double helices, with azimuthal symmetries (wavenumbers) $m=-1$ ($m_{1}$) and $-2$ ($m_{2}$). Such a new high-helical quasi-periodic regime is clearly observable at the phase diagrams depicted in figure 3(b,c), and is associated to stronger pressure fluctuations (see figure 4e), whose loci are less aligned with the flow streamlines, on account of the larger radial inclination of spiral threads.

Finally, for $\unicode[STIX]{x1D6FA}\geqslant 2.4$, the wake becomes irregular again, and the higher swirl of the flow induces stronger double-threaded helices in the near wake that tilt and connect to form vortex loops in different directions, as observed in figure 2 for $\unicode[STIX]{x1D6FA}=3$.

Such vortex loops rotate with the vortical structure as they move downstream, changing progressively their orientation, before dissipating in the far wake. The irregular regime presents uneven time histories of force coefficients, as displayed in figure 3(a), where an important modulation of the amplitude and multiple frequencies are distinguishable in $c_{x}$ and $c_{y}$. As a consequence, the phase diagram of lift components $(c_{y},c_{z})$ becomes entangled (figure 3b). Interestingly, the radial extension of such a phase portrait is smaller than that corresponding to the irregular regime reported for $\unicode[STIX]{x1D6FA}=1.6$, and so is the largest fluctuating value of the total lift, as shown in figure 3(c). In addition, the $(c_{l},c_{x})$ diagrams also show that the fluctuations of $c_{x}$ become progressively larger within this regime as $\unicode[STIX]{x1D6FA}$ increases, leading to wider phase diagrams. This is an indication of the fostering of the shedding process, associated to important overall pressure changes in the rear part of the sphere, as figure 4(f) depicts. There it can also be estimated that the time-averaged separation angle, $\unicode[STIX]{x1D6F7}_{s}$, decreases with growing rotation since it acts by lowering the pressure on the equator, thus creating an increasingly adverse gradient that fosters the flow separation.

Figure 5.

Fast Fourier transform spectra, listing the main frequency components, for different rotation parameters, $\unicode[STIX]{x1D6FA}$, at $Re=250$ corresponding to the (a) $y$-axis lift coefficient, $c_{y}^{\prime }=c_{y}-C_{y}$, (b) drag coefficient, $c_{x}^{\prime }=c_{x}-C_{x}$, and the (c) azimuthal velocity $w^{\prime }=w-W$ at $(x,y,z)=(2,0.5,0)$. The same amplitude logarithmic scale is used for all fast Fourier transform spectra, including decades that range from $10^{0}$ to $10^{-6}$.

3.2 Frequency analysis

We next further analyse the flow regimes and transitions in terms of frequency components from spectra obtained by computing the fast Fourier transform (FFT) of time fluctuating values of force coefficients, $c_{y}^{\prime }=c_{y}-C_{y}$ and $c_{x}^{\prime }=c_{x}-C_{x}$, and the azimuthal velocity, $w^{\prime }=w-W$, at the selected location $(x,y,z)=(2,0.5,0)$, which displays the largest amplitudes of fluctuating components (not shown) in comparison with other axial and radial positions. Corresponding FFT spectra of $c_{y}^{\prime }$, $c_{x}^{\prime }$ and $w^{\prime }$ for selected values of $\unicode[STIX]{x1D6FA}$ are included in figure 5, where the main components of frequency are identified and labelled to enlighten (and simplify) the discussion on the flow transitions. Note also that the same logarithmic scale is used for all FFT spectra aiming at establishing a quantitative comparison of amplitudes associated to different frequency components.

For the non-rotating sphere ($\unicode[STIX]{x1D6FA}=0$), the wake is steady and, therefore, no FFT spectrum is included. At $\unicode[STIX]{x1D6FA}=0.4$, the signal of $c_{y}$, shown in figure 3(a), is periodic according to its FFT, with a fundamental frequency $f_{a}$, which is related to the rate of rotation of the wake. However, the harmonic $2f_{a}$ develops at the near wake, as observed in the spectrum of $w^{\prime }$, although its magnitude is weak in comparison with the main peak. Interestingly, if the sphere’s rotation frequency is denoted as $f_{r}$, the ratio $f_{a}/f_{r}\simeq \frac{1}{3}$, which agrees well with previously reported values by Pier (Reference Pier2013) and Skarysz et al. (Reference Skarysz, Rokicki, Goujon-Durand and Wesfreid2018) for low $\unicode[STIX]{x1D6FA}$.

At $\unicode[STIX]{x1D6FA}=0.6$, a new frequency, labelled $f_{b}$, emerges in the spectrum of $c_{x}^{\prime }$ (figure 5b). As a consequence, new peaks resulting from the interaction of these two incommensurate frequencies $f_{a}$ and $f_{b}$, arise in the spectra of $c_{y}^{\prime }$ and $w^{\prime }$ (figure 5a,c), leading to the previously described quasi-periodic regime. In particular, modulations in the amplitude of $c_{y}(t)$ in figure 3(a) and the orbit of the phase diagram $(c_{y},c_{x})$ in figure 3(b) occur at a frequency $f_{a}+f_{b}$, although they are barely discernible on account of the low magnitude of such frequency components shown in the FFT of $c_{y}^{\prime }$.

As the rotation rate $\unicode[STIX]{x1D6FA}$ increases, harmonics and nonlinear interactions between the previous frequencies produce new peaks of moderate amplitude (see, e.g. $f_{b}-f_{a}$, $f_{a}+f_{b}$, $2f_{b}+f_{a}$ or $2f_{b}-f_{a}$ in the spectra for $\unicode[STIX]{x1D6FA}=1.2$ and 1.4). The frequency $f_{a}$ increases with $\unicode[STIX]{x1D6FA}$ which indicates an acceleration of the wake rotation, although such growth becomes progressively less important within the quasi-periodic regime, leading to a rate $f_{a}/f_{r}$ that decreases monotonously.

As described earlier, this quasi-periodic regime prevails at the wake until $\unicode[STIX]{x1D6FA}=1.6$, when the high-helical irregular regime sets in (see figure 2). The frequency analysis shows that, at $\unicode[STIX]{x1D6FA}=1.6$, some of the former fundamental frequency components and nonlinear interactions are distinguishable in the spectra of $c_{y}^{\prime }$ and $w^{\prime }$ as broader peaks (no clear peaks are observed for the $c_{x}^{\prime }$ spectrum), being an indication of richer dynamics, compatible with a chaotic wake behaviour (an observation which will be further discussed in § 7). However, the most interesting feature is the development of a third incommensurate frequency component, denoted $f_{c}$, that presents the largest amplitude at the wake, so that an irregular regime is related to the confluence of three incommensurate frequencies.

In addition, the combined frequency $2f_{c}-f_{b}$ stands out in the $w^{\prime }$ spectrum (figure 5c), which, as it will be detailed in the following section, is associated to the swirl of a double spiral structure which develops in the near wake.

Figure 6.

Fast Fourier transform spectra of axial and transverse velocities, $u^{\prime }=u-U$ and $w^{\prime }=w-W$, and pressure, $p^{\prime }=p-P$, at downstream axial locations, $x\leqslant 10$, and radial locations $r=[0,0.5]$ for the selected value of $\unicode[STIX]{x1D6FA}=1.6$ and $Re=250$.

The irregular state evolves as $\unicode[STIX]{x1D6FA}$ increases, until $\unicode[STIX]{x1D6FA}=2.2$, where spectra display a new quasi-periodic behaviour related to a synchronization regime, dominated by frequency $f_{c}$ and its quadratic interactions with $f_{b}$ (see figure 5a). In particular, modulations in amplitude of $c_{y}^{\prime }(t)$ in figure 3(a) (and in the orbit of the lift diagram) take place with a frequency $f_{c}-f_{b}$. In addition, the amplitude of peaks is considerably smaller for $c_{x}^{\prime }$, indicating weaker fluctuating dynamics, as observed in the $(c_{y},c_{x})$ diagram (figure 3c). Interestingly, at the near wake, the dynamics is governed by $2f_{c}-f_{b}$, as displayed in the $w^{\prime }$ spectrum (figure 5c).

Finally, for $\unicode[STIX]{x1D6FA}\geqslant 2.4$, the same dominating frequency components are present in the spectra, although now multiple additional peaks develop, all of them featuring broader bases. The latter indicates again a transition towards the irregular regime.

To analyse the axial extent, spatial prevalence and to infer the azimuthal nature of the different features associated with the different frequencies in the near wake, we display in figure 6 the FFT spectra of axial and transverse velocities, $u^{\prime }=u-U$ and $w^{\prime }=w-W$, and pressure, $p^{\prime }=p-P$, at downstream axial, $x\leqslant 10$, and radial locations $r=[0,0.5]$ for the selected value of $\unicode[STIX]{x1D6FA}=1.6$, where the three incommensurate frequencies $f_{a}$, $f_{b}$ and $f_{c}$, and the derived frequency $2f_{c}-f_{b}$ are present. The energy of such frequencies decays in general for $x\geqslant 6$, and peaks associated to $f_{c}$ and $2f_{c}-f_{b}$ are particularly energetic for all variables. In particular, $2f_{c}-f_{b}$ stands out in FFT spectra taken at $r=0.5$ and $x\leqslant 4$, indicating that the associated structures are mainly near wake structures. The frequency $f_{c}$ features the largest spectral magnitude at $r=0.5$ and $r=0$ for the transverse velocity, congruent with a helical pattern extending up to $x=10$. A similar picture can be observed for the component $f_{a}$, associated to the rotating helical structures stemming from the periodic regime, however, the corresponding peaks are less energetic at this level of $\unicode[STIX]{x1D6FA}$. Finally, the peak $f_{b}$ is only observable at $r=0$ for spectra of the pressure $p^{\prime }$ and axial velocity $u^{\prime }$, extending until $x=10$. Thus, the associated phenomenon mostly occurs at a centred location, as shown by the shedding in figure 2 for $\unicode[STIX]{x1D6FA}=1$.

3.3 Flow decomposition

3.3.1 Higher-order dynamic mode decomposition

At this point, a deeper analysis of vortical structures is required to better identify the wake dynamics and nature of the different flow regimes. To that aim, flow decomposition techniques will be employed in order to infer the spatial symmetries that dominate each regime, and structures associated to each component frequency. First, a higher-order dynamic mode decomposition (HODMD) technique (Le Clainche & Vega Reference Le Clainche and Vega2017a) has been employed to extract dynamic modes linked to particular frequencies observed in figure 5 (see appendix B for details). The analysis is performed with snapshots of the velocity field taken at $x=2$, $\boldsymbol{u}(x=2,y,z,t)$, where the largest spectral amplitudes are found, as shown in figure 6.

Results for $Re=250$ are summarized in figure 7, where spatial structures are identified by means of contours of the real part of the normalized axial vorticity $\mathbb{R}(\unicode[STIX]{x1D71B}_{x})$ (using the mode infinity norm $\Vert \boldsymbol{u}\Vert _{\infty }$), along with their associated frequency, for selected values of the rotation parameter $\unicode[STIX]{x1D6FA}$. To simplify the analysis, only characteristic modes featuring the largest amplitudes are included (an example of the spectral complexity obtained can be found in appendix B).

Figure 7.

Contours of the normalized axial vorticity $\unicode[STIX]{x1D71B}_{x}$ (real part) for the most representative modes identified through HODMD analysis for selected values of $\unicode[STIX]{x1D6FA}$ at $Re=250$. Modes are arranged from left to right in increasing order of azimuthal wavenumber.

Although not shown, the non-rotating sphere’s wake is characterized by a steady (frequency $f=0$) mode displaying a dipole of axial vorticity, compatible with an azimuthal symmetry with wavenumber $m=1$ ($m_{1}$). With rotation, the mode becomes unsteady and a periodic regime emerges in the wake.

At $\unicode[STIX]{x1D6FA}=0.4$, the mode presents a frequency $f(m_{1})=0.043$ which corresponds to the peak $f_{a}$ identified in figure 5(c). As $\unicode[STIX]{x1D6FA}$ increases up to a value of 0.6, a new fundamental frequency $f_{b}$ was identified in the spectra of figure 5 and related to the periodic shedding of vortices, leading to a new unsteady quasi-periodic regime in the wake. When the dynamic modes are computed, aside from the former $m_{1}$ structure associated to $f=0.061\equiv f_{a}$, a new energetic mode is found for $f(m_{0})=0.082$. This frequency corresponds to the peak $f_{b}$ of figure 5, which identifies the periodic shedding observable in figure 2 at $\unicode[STIX]{x1D6FA}=1$. Interestingly, the mode shows a nearly axisymmetric distribution of vorticity, featuring an outer ring of positive vorticity and an inner core of negative vorticity. Such topology might be related to an axisymmetric pulsation from the excitation of a $m=0$ mode that leads to the inner shedding of vortices, in line with the observations made from figure 2.

For $\unicode[STIX]{x1D6FA}=1.6$, a new irregular regime sets in the wake, characterized by a double-helical flow structure, as displayed in figure 2. This double-threaded wake may be an indication of the destabilization of a mode with azimuthal wavenumber $m=2$ ($m_{2}$), as occurs for generic swirling flows at high values of flow rotation (see, e.g. Ruith et al. Reference Ruith, Chen, Meiburg and Maxworthy2003). In particular, the HODMD analysis reveals that the most energetic mode corresponds to an $m_{1}$ symmetry, which is qualitatively different to the dipole identified at lower rates of rotation, and resembles the Yin-Yang mode (Auguste, Fabre & Magnaudet Reference Auguste, Fabre and Magnaudet2010) identified experimentally by Skarysz et al. (Reference Skarysz, Rokicki, Goujon-Durand and Wesfreid2018), its frequency being $f(m_{1})\equiv f_{c}=0.205$. Additionally, the second largest amplitude is related to $f=0.262$, which corresponds to the frequency component $2f_{c}-f_{b}$ identified in figure 5, and displayed a quadrupole of vorticity compatible with an $m=2$ symmetry. Such azimuthal symmetry is the outcome of the interaction of harmonic $2f_{c}$ ($2\cdot m_{1}$) and $f_{b}$ ($m_{0}$), and would be responsible for the double-helical structure observed in figure 2. Moreover, as in the previous quasi-periodic regime, the modes associated to $f_{a}=0.093$ ($m_{1}$ symmetry) and $f_{b}=0.141$ ($m_{0}$ symmetry) are also identified at the wake, although their corresponding growth rates are small in comparison with those of $f_{c}$ and $2f_{c}-f_{b}$, which mainly govern the flow dynamics.

At $\unicode[STIX]{x1D6FA}=2.2$, a more organized wake state appears, through a synchronization of the wake at selected frequencies $2f_{c}-f_{b}$, $f_{c}$ and $f_{b}$, and the corresponding modes display a better spatial resolution. The same picture is obtained for $\unicode[STIX]{x1D6FA}>2.2$ within the irregular regime, although modes feature a more irregular topology.

A summary of the evolution with $\unicode[STIX]{x1D6FA}$ of frequencies associated to previously described most energetic modes is shown in figure 8(a). As observed, the wake rotation frequency $f_{a}$ grows at low rotation rates to reach a constant value at $\unicode[STIX]{x1D6FA}\simeq 1.2$. The shedding frequency $f_{b}$ emerges around $\unicode[STIX]{x1D6FA}=0.6$, causing the transition towards the quasi-periodic regime, and grows monotonously with $\unicode[STIX]{x1D6FA}$ within the range investigated. The abrupt transition to the irregular regime at $\unicode[STIX]{x1D6FA}=1.6$ is produced by the existence of a new fundamental frequency component, $f_{c}$, and its quadratic interaction $2f_{c}-f_{b}$, as sketched in figure 8(a), where the trend has also been plotted for the sake of comparison. As expected, the dynamics is sped up with rotation, giving growing trends of $f_{b}$ and $f_{c}$ within the range $1.6\leqslant \unicode[STIX]{x1D6FA}\leqslant 3$. On the other hand, the apparent mode azimuthal wavenumber $m$ is also listed for each frequency component in figure 8(a) to qualitatively indicate the azimuthal symmetries present for each regime. Below $\unicode[STIX]{x1D6FA}=1.6$ the flow is mainly governed by an $m_{1}$ symmetry, while above this threshold there is a coexistence of $m_{1}$ and $m_{2}$ structures.

Figure 8.

Flow decomposition at $Re=250$: evolution with $\unicode[STIX]{x1D6FA}$ of (a) characteristic frequencies corresponding to dominant modes; (b) enstrophy, $\unicode[STIX]{x1D700}(m)$, for the most energetic azimuthal symmetries (FAD); and (c) relative contribution to total enstrophy for symmetries $m=0,1,2$ and 3 (FAD). The vertical lines in (b,c) distinguish regimes with different dominant azimuthal symmetries.

4.1 Qualitative description of flow regimes

The different qualitative wake regimes encountered for $Re=500$ and increasing $\unicode[STIX]{x1D6FA}$ are summarized in figure 9, where instantaneous plots of vortical structures are displayed using isosurfaces of the $Q$-quantity, and in figure 10, which includes time histories of $c_{x}(t)$ and $c_{y}(t)$, and corresponding phase diagrams $(c_{y},c_{z})$ and $(c_{l},c_{x})$ for selected values of $\unicode[STIX]{x1D6FA}$. Note that, unlike $Re=250$, a level of $Q=10^{-2}$ has been set in figure 9 since a higher threshold is required to display coherent and neat topologies (the larger axial shear induced by the stronger axial velocity gradients increases the magnitude of the rotation tensor and then larger values of $Q$ are found at the wake as $Re$ grows).

As reported by Tomboulides & Orszag (Reference Tomboulides and Orszag2000), at $\unicode[STIX]{x1D6FA}=0$ the wake is characterized by an irregular hairpin vortex-shedding, featuring an uneven azimuthal rotation of the separation point at the sphere’s rear. Thus, the hairpin vortices observed in figure 9 display different orientations at the wake and there is no trace of the planar symmetry seen at $Re=250$. The irregular nature is also observed in the time histories of $c_{y}(t)$ and $c_{x}(t)$ (see figure 10a), and corresponding phase diagrams $(c_{y},c_{z})$ and $(c_{l},c_{x})$ in figure 10(b,c).

With rotation, the irregular shedding regime remains present up to $\unicode[STIX]{x1D6FA}=0.4$, where a sudden transition towards a periodic regime takes place. As observed in figure 9, the wake arranges itself into a coherent helical structure displaying two counter-rotating threads (configuring the outer spiral) and an inner near wake spiral of negative axial vorticity, which, in fact, is attached to the main negative thread. Such structures rotate together in a frozen manner, which translates into a steady drag time history, $c_{x}(t)$, and periodic lift components, $c_{y}(t)$, as displayed in figure 10(a). As a consequence, the phase diagram of $(c_{y},c_{z})$ is represented by a perfect circular orbit (see figure 10b), while the corresponding phase diagram $(c_{l},c_{x})$ is a point (see figure 10c). Interestingly, the more organized nature of the wake and the constant eccentricity of near wake structures leads to a smaller value of the mean total lift than that associated to the irregular shedding regime, as seen in figure 10(c). Such a regime is equivalent to the high-frequency or spiral frozen regimes respectively identified by Pier (Reference Pier2013) or Jiménez-González et al. (Reference Jiménez-González, Sanmiguel-Rojas, Sevilla and Martínez-Bazán2013).

The spiral frozen regime is soon disrupted when $\unicode[STIX]{x1D6FA}$ increases above 0.6. In particular, the inner near wake spiral elongates and is torn apart downstream due to the induced shear from the outer double spiral structure, which now becomes less coherent, as observed for $\unicode[STIX]{x1D6FA}=0.8$ (figure 9). As shown in figure 10, such a wake regime is characterized by an oscillating time history of $c_{x}(t)$ and low frequency modulations in the lift $y$-component signal, $c_{y}(t)$, which is no longer periodic. Thus, the phase diagram $(c_{y},c_{z})$ presents a quasi-periodic orbit, which leads to a fluctuating behaviour of the total lift $c_{l}(t)$ (see figure 10c).

At $\unicode[STIX]{x1D6FA}=1.1$ the wake displays a more irregular structure, as observed in figure 9, and the shedding of vortices of smaller scale from the inner vortical structure is strengthened. In addition, the outer double spiral is less clear at the wake, since the high level of rotation forces the spirals to tilt and connect, leading to the formation of incipient loops and the presence of multiple characteristic scales. The $y$-axis lift time history $c_{y}(t)$ displays a stronger, irregular modulation of low frequency in figure 10(a), and larger fluctuations in the drag coefficient $c_{x}(t)$. As a consequence, the phase diagrams of $(c_{y},c_{z})$ and $(c_{l},c_{x})$ display erratic quasi-periodic orbits (see figure 10b insets).

However, within the range $1.1\leqslant \unicode[STIX]{x1D6FA}\leqslant 1.3$, the flow presents alternative periods of the previously described spiral configuration and an irregular topology (respectively denoted L and H states hereafter), characterized by the definitive disruption of vortical spiral threads that connect creating vortex loops in the near wake (see $\unicode[STIX]{x1D6FA}=1.1$ H in figure 9), while the whole flow rotates. Interestingly, traces of double spiral structures and multiple scales are visible at the near wake, suggesting the existence of energetic modes of symmetry $m\geqslant 2$. A deeper analysis of the time history of the drag coefficient (see figure 10d) reveals long time bistable dynamics, characterized by the random switching of $c_{x}(t)$ between two different average values, respectively denoted as high-drag (H) and low-drag (L). The events of low- and high-drag respectively correspond to the prevalence of the modulated quasi-periodic spiral configuration and this new irregular regime in the flow (see wake structures displayed for $\unicode[STIX]{x1D6FA}=1.1$ L and $\unicode[STIX]{x1D6FA}=1.1$ H in figure 9).

The existence of two flow states associated to respective high and low values of drag is clearly shown in figure 10(e), where contours of the probability density function (PDF) of $c_{x}^{\prime }=c_{x}-C_{x}$ (normalized by the most probable value) are presented for each value of the rotation rate $\unicode[STIX]{x1D6FA}$. As observed, the level of fluctuations for the periodic and quasi-periodic regimes for $\unicode[STIX]{x1D6FA}<1.1$ is small and centred around zero; however, for $1.1\leqslant \unicode[STIX]{x1D6FA}\leqslant 1.3$, there are two peaks with a high probability of occurrence around $c_{x}^{\prime }=0.1$ and $c_{x}^{\prime }=-0.1$. The differences between both states in terms of force coefficients and phase diagrams are shown in figure 10(a–c) for $\unicode[STIX]{x1D6FA}=1.1$, where conditional time series are used for the plots (i.e. $c_{x}^{\prime }(t)>0.05$ and $c_{x}^{\prime }(t)<-0.05$ for H and L states, respectively). There, it is evidenced the more irregular nature of the high-drag state, which also features larger values of mean and fluctuating lift. In addition, the switching time is very short in comparison with the long time interval of both states (figure 10d), and the number of switches between states increases with $\unicode[STIX]{x1D6FA}$. However, at $\unicode[STIX]{x1D6FA}=1.3$, the H state is soon reversed when established at the wake, leading to a lower probability of occurrence.

For $\unicode[STIX]{x1D6FA}\geqslant 1.4$, the flow is no longer bistable, as shown in figure 10(e). In fact, the lower branch associated to the L regime seems to govern the dynamics as $\unicode[STIX]{x1D6FA}$ increases. As seen, within the range $1.4\leqslant \unicode[STIX]{x1D6FA}\leqslant 3$, the time history of $c_{x}$ features large-amplitude fluctuations, which anticipate the existence of a highly irregular regime. In particular, at $\unicode[STIX]{x1D6FA}=3$ the wake is characterized by the rotation and massive shedding of vortex loops (see figure 9), which are produced at the shear layer and distribute downstream following the helical pattern described earlier.

Such a scenario leads to larger fluctuations of lift components, $c_{y}$ and $c_{z}$, and drag, $c_{x}$, and complex trajectories in the phase diagrams of figure 10(a–c). The differences in topology between regimes are further illustrated in figure 11 by means of contours of standard fluctuations of the pressure coefficient, $C_{p}^{\prime }$, and time-averaged velocity streamlines at $y=0$. The irregular vortex-shedding regime at $\unicode[STIX]{x1D6FA}=0$ displays a higher level of fluctuations along the shear layers, whereas for the periodic and quasi-periodic regimes for $\unicode[STIX]{x1D6FA}\leqslant 1.3$ (i.e. low-drag behaviour), the largest magnitude of $C_{p}^{\prime }$ is distributed at the cores of recirculating bubble and downstream of it. In general, the magnitude of the pressure fluctuations increases with $\unicode[STIX]{x1D6FA}$, while the recirculating cores move towards outer radial locations. The picture differs for the high-drag regime (see $\unicode[STIX]{x1D6FA}=1.2$ H in figure 11), whose wake is characterized by a double-celled recirculating bubble.

Figure 11.

Contours of fluctuations of the pressure coefficient, $C_{p}^{\prime }$, along with time-averaged velocity streamlines at $y=0$ for selected values of the rotation parameter $\unicode[STIX]{x1D6FA}$ at $Re=500$.

Note that similar double-celled structures were identified by Ruith et al. (Reference Ruith, Chen, Meiburg and Maxworthy2003) for swirling flows with wake-like axial velocity profiles at intermediate values of the rotation rate and Reynolds number. Moreover, the near wake displays intense fluctuations downstream of the recirculating bubble, which is now abruptly shortened.

Finally, the irregular shedding regime observed at large values of the rotation rates (see, e.g. $\unicode[STIX]{x1D6FA}=3$) displays again a single vortex ring in the near wake. A wide region of high magnitude of $C_{p}^{\prime }$ is found at the axis and rear end of the recirculating bubble, where the fluctuations are more intense.

4.2 Frequency analysis

In figure 12 we show the FFT spectra of fluctuating values of forces $c_{y}^{\prime }(t)$ and $c_{x}^{\prime }(t)$, and the azimuthal velocity $w^{\prime }(t)$ taken at $(x,y,z)=(2,0.5,0)$ for selected values of the rotation rate $\unicode[STIX]{x1D6FA}$. To follow the same approach as in § 3, fundamental dominant frequencies will be identified according to their different nature, so that $f_{a}$ is related to wake threads rotation, $f_{b}$ to shedding or $f_{c}$ for secondary helical dynamics.

Figure 12.

Fast Fourier transform spectra, listing main frequency components, for different rotation parameters, $\unicode[STIX]{x1D6FA}$, at $Re=500$, corresponding to the (a) $y$-axis lift coefficient, $c_{y}^{\prime }=c_{y}-C_{y}$, (b)  drag coefficient, $c_{x}^{\prime }=c_{x}-C_{x}$, and the (c) azimuthal velocity $w^{\prime }=w-W$ at $(x,y,z)=(2,0.5,0)$. The computation of FFT spectra at $\unicode[STIX]{x1D6FA}=1.1$ for low- and high-drag regimes have been performed using conditional time series.

First, the spectra associated to the irregular regime at $\unicode[STIX]{x1D6FA}=0$ display broad frequency components, with two main peaks at $f_{s}=0.163$ and $f_{i}=0.049$, which are respectively associated to the vortex shedding ($f_{s}$) and the irregular azimuthal pulsation of the shedding point at the sphere’s rear ($f_{i}$). In particular, the peaks of the shedding process are neater for $c_{x}^{\prime }$ and $w^{\prime }$, while the lift $c_{y}^{\prime }$ seems to be dominated by the combination of previous main frequencies $f_{s}-f_{i}$.

As the rotation is applied, the wake organizes itself, and as observed in the FFT spectra of $c_{y}^{\prime }$ and $w^{\prime }$, the dynamics is mainly dominated by the single frequency, $f_{a}=f_{s}-f_{i}$, which corresponds to the rate of rotation of the vortical structures depicted in figure 9 (harmonics $2f_{a}$ and $3f_{a}$ are also identified in the FFT spectra of $w^{\prime }$). Thus, such a regime emerges after a wake synchronization at the combination of former shedding process frequencies, $f_{s}-f_{i}$. The frozen behaviour renders the flow steady from a reference system rotating with the wake and, consequently, the FFT spectra f $c_{x}^{\prime }$ does not display any significant peak in figure 12. Note that, unlike the frozen state reported at $Re=250$, now the rate is $f_{a}/f_{r}>1$, indicating that the wake rotates faster than the sphere.

A slight increase of the rotation rate up to 0.6 leads to the emergence of a new incommensurate frequency peak $f_{b}$ in the FFT spectra of drag $c_{x}^{\prime }$, which may be related to the vortex shedding from the internal near wake spiral identified earlier. Additionally, a secondary peak emerges in the FFT spectra of $c_{y}^{\prime }$, that is, the nonlinear combination of both wake rotation and shedding frequencies, i.e. $f_{a}-f_{b}$, thus leading to the quasi-periodic time series $c_{y}(t)$ and phase portraits $(c_{y},c_{z})$ of figure 10. Moreover, the wake also displays harmonic and derived frequencies downstream (see the FFT spectra of $w^{\prime }$).

The energy content of such frequencies rises as the rotation increases up to $\unicode[STIX]{x1D6FA}=1.1$ for the L regime, where the spectra display broader features, indicating more complex wake dynamics, although the same coherent peaks are still mostly recognizable. However, a new incommensurate frequency $f_{c}$ is observed in the FFT spectra of drag $c_{x}^{\prime }$. This tertiary low-frequency component may represent a modulation of the shedding process governed by $f_{b}$, considering the existence of nonlinear interactions $f_{b}-f_{c}$ and $f_{b}+f_{c}$ in the spectrum. Then, such low-frequency pulsation seems to be the precursor for the sudden topological change observed in the near wake of the H regime, whereby the two-celled recirculating bubble appears (figure 11). In particular, the different dynamics obtained at $\unicode[STIX]{x1D6FA}=1.1$ for the H regime is mainly governed by the combined frequency $f=2f_{b}-f_{c}$ and its harmonic $2f=2(2f_{b}-f_{c})$, in view of the FFT spectra of $w^{\prime }$ (figure 12), being the latter frequency associated to the existence of energetic $m=2$ structures at the wake (figure 9), as will be shown below. Other combinations of previously reported frequencies are visible in the FFT spectra of $c_{y}^{\prime }$ and $c_{x}^{\prime }$, e.g. the nonlinear components $f_{a}-f_{b}$ and $(2f_{b}-f_{c})+(f_{a}-f_{b})=f_{a}+f_{b}-f_{c}$. The peaks display generally broader bases, indicating more irregular dynamics than that of the L regime (as in the phase diagrams in figure 10b,c). Thus, the reported H regime seems to arise as a process of wake synchronization at a derived frequency from the two incommensurate values $f_{b}$ and $f_{c}$. As generally occurs in bistable flows (see e.g. Cadot, Evrard & Pastur Reference Cadot, Evrard and Pastur2015), the random switching between both wake states might be triggered by any flow perturbation that selects predominantly one of the states.

Finally, for $\unicode[STIX]{x1D6FA}>1.4$, the wake becomes more irregular, being the dominant frequencies $f_{a}$ and the interaction $f_{a}-f_{b}$, indicating that the wake dynamics is mainly governed by the low-drag regime frequency features. For such rotation rates, all the displayed FFT spectra are characterized by noisy, broadband peaks, indicating a potential transition to chaos. As in § 3, to analyse the spatial prevalence of the frequencies, we display in figure 13 the FFT spectra of axial and transverse velocities, $u$ and $w^{\prime }$, and pressure, $p^{\prime }$, at $x\leqslant 10$ and $r=[0,0.5]$ for $\unicode[STIX]{x1D6FA}=1$, where the three incommensurate frequencies $f_{a}$, $f_{b}$ and $f_{c}$ are present. In general, the largest spectral magnitudes are found at $x=2$. In addition, it can be observed that the frequency $f_{a}$ displays energetic peaks in the FFT spectra computed at $r=0.5$ for all variables and prevails at least until $x=10$. The fundamental frequency component $f_{b}$ is particularly energetic in $r=0$ for the axial velocity $u^{\prime }$ and pressure $p^{\prime }$, which suggests that the associated phenomenon takes place at a centred location, as the shedding process from the inner spiral shown in figure 9. Additionally, the low-frequency peak $f_{c}$ arises in the spectra of the axial velocity, $u^{\prime }$, and pressure, $p^{\prime }$, at $r=0$ and the near wake for $x<4$, so that it might be related to a pulsation of the recirculating bubble, that triggers the sudden transition to the H regime at $\unicode[STIX]{x1D6FA}=1.1$, through the activation of the reported frequency component $2f_{b}-f_{c}$. Finally, the combined frequency $f_{a}-f_{b}$ stands out as an especially intense peak in the FFT spectrum of $p^{\prime }$ at $r=0.5$, and observable for $w^{\prime }$ at both radial locations, but limited to $x\leqslant 6$, which is compatible with a near wake helical structure, as will be shown below.

Figure 13.

Fast Fourier transform spectra of the axial and transverse velocities, $u^{\prime }=u-U$ and $w^{\prime }=w-W$, and pressure, $p^{\prime }=p-P$, at downstream axial locations, $(x\leqslant 10)$, and radial locations $r=[0,0.5]$ for the selected value of $\unicode[STIX]{x1D6FA}=1$ and $Re=500$.

Figure 14.

Contours of the axial vorticity $\unicode[STIX]{x1D71B}_{x}$ (real part), for the most energetic modes identified through HODMD analysis at $x=2$, for selected values of $\unicode[STIX]{x1D6FA}$ at $Re=500$. Modes are arranged from left to right in increasing order of azimuthal wavenumber.

4.3 Flow decomposition

4.3.1 Higher-order dynamic mode decomposition

To conclude the discussion on the flow regimes for $Re=500$, we apply again the HODMD analysis at $x=2$. Results are summarized in figure 14, where contours of the normalized axial vorticity $\unicode[STIX]{x1D71B}_{x}$, corresponding to dominant modes with the largest growth rate, are displayed for selected values of the rotation rate $\unicode[STIX]{x1D6FA}$. Without rotation ($\unicode[STIX]{x1D6FA}=0$), the wake is dominated by two fundamental frequencies, $f(m_{1})=0.163$ and $f(m_{0})=0.049$, which respectively correspond to the shedding frequency, $f_{s}$, and the irregular azimuthal rotation rate of the shedding location, $f_{i}$, identified in the spectrum of $c_{x}^{\prime }$ in figure 12. The vortex shedding mode presents an azimuthal symmetry $m_{1}$, with two counter-rotating vortices, while the irregular rotation of the shedding location explores approximately the whole azimuth at the rear of the sphere, so that the mode can be considered statistically axisymmetric (note that the slow dynamics of such phenomenon would require a very long time history of snapshots to fully retrieve the statistical axisymmetry).

For $\unicode[STIX]{x1D6FA}=0.4$, the wake was shown to display periodic dynamics that is retrieved at the flow decomposition, which only identifies an unstable mode with frequency $f_{m_{1}}=0.143\equiv f_{a}$, related to a dipole of axial vorticity close to the axis and an induced vorticity around them. Such periodic dynamics is broken as $\unicode[STIX]{x1D6FA}$ grows, as seen, for instance, when $\unicode[STIX]{x1D6FA}=0.8$, due to the occurrence of a nearly axisymemtric shedding of frequency $f_{m_{0}}=0.133\equiv f_{b}$. This mode presents a negative vorticity of high intensity at the axis, coinciding with the location of the near wake spiral (see figure 9), from where vortices are detached. Additionally, HODMD identifies secondary frequency components in the form $f=n_{1}f_{a}\pm n_{2}f_{b}$, with $n_{1}$ and $n_{2}$ being natural numbers. As an instance, the energetic mode with $f=0.368$, which correspond to $3f_{a}-f_{b}$ in figure 12 is displayed. As expected from the simple interaction of frequencies, the structures show an $m_{3}$ symmetry.

The scenario changes at $\unicode[STIX]{x1D6FA}=1.1$, where the quasi-periodic dynamic is strongly modulated by the new frequency component $f_{c}$, which is mainly identified in the FFT spectra of $c_{x}^{\prime }$ and $p^{\prime }$ in the near wake. According to the decomposition, a new mode of frequency $f=0.033\equiv f_{c}$ and moderate amplitude is present at the flow, and is characterized by a helical structure featuring an entangled spiral dipole of vorticity. Thus, it constitutes a wake modulation phenomenon with an $m_{1}$ symmetry. In addition, the complexity of the wake at the L regime is also the outcome of the strengthening of derived frequencies, as the $f_{a}-f_{b}=0.046$, whose helical structure presents an $m_{1}$ symmetry as expected from the interaction between previously described modes $f_{a}$ and $f_{b}$.

The analysis of the wake corresponding to the high-drag regime clearly identifies the energetic frequency components $f=0.253$ and its harmonic $2f=0.505$, which correspond to the peaks of $2f_{b}-f_{c}$ and $2(2f_{b}-f_{c})$ previously observed in the FFT analysis. The interaction between fundamental frequencies entails that both modes present respectively helical $m_{1}$ and $m_{2}$ azimuthal symmetries. The elongated contours featured by the component $2f_{b}-f_{c}$ anticipates the existence of the intense spiral topology and incipient vortex loops. Interestingly, the modes do not present vorticity close to the axis, which can be associated to the existence of the double-celled recirculating region in the near wake.

Finally, the irregular regime established for $\unicode[STIX]{x1D6FA}>1.4$ is fundamentally characterized by $m_{1}$ structures associated to the frequency component $f_{a}$, that now presents a larger radial extension than in the precedent regime. In particular, unlike the previous modes governing the wake dynamics for the H regime, the structures cover the vicinity of the axis and are more elongated, indicating a tilting of threads that leads to the connection and formation of vortex loop, as displayed in figure 9.

Figure 15.

Flow decomposition at $Re=500$: evolution with $\unicode[STIX]{x1D6FA}$ of the (a) characteristic frequencies corresponding to dominant modes; (b) enstrophy, $\unicode[STIX]{x1D700}(m)$, for the most energetic azimuthal symmetries (FAD); and the (c) relative contribution to the total enstrophy for symmetries $m=0,1,2$ and 3 (FAD). The vertical lines in (b,c) distinguish regimes with different dominant azimuthal symmetries. Black and white filled symbols in (a) respectively represent fundamental and combined frequencies; while grey symbols in (b,c) stand for values obtained at the conditional high-drag regime.

A summary of the evolution of the dominant frequencies, associated to the most energetic modes at the wake, is included in figure 15(a). As observed, the irregular shedding regime at low $\unicode[STIX]{x1D6FA}$ is governed by both the shedding frequency, $f_{s}$, and the azimuthal rotation of the shedding point related to $f_{i}$. The combined frequency $f_{s}-f_{i}$ grows linearly with $\unicode[STIX]{x1D6FA}$ and becomes dominant at $\unicode[STIX]{x1D6FA}=0.4$, where the wake becomes frozen and rotates periodically at the synchronization frequency $f_{a}=f_{s}-f_{i}$. The shedding frequency $f_{b}$ appears at $\unicode[STIX]{x1D6FA}=0.6$, rendering the wake dynamics quasi-periodic. Both $f_{b}$ and the nonlinear combination $f_{a}-f_{b}$ grow monotonously with $\unicode[STIX]{x1D6FA}$ up to 1.1 approximately. The destabilization of the frequency $f_{c}$ around $\unicode[STIX]{x1D6FA}=1$ implies that the modulation of quasi-periodic dynamics reported for the low-drag regime, within the range $1.1\leqslant \unicode[STIX]{x1D6FA}\leqslant 1.3$, while its quadratic interaction with $f_{b}$, i.e. $2f_{b}-f_{c}$, governs the dynamics of the high-drag regime together with the secondary frequencies $2(2f_{b}-f_{c})$ (of $m_{2}$ symmetry) and $f_{a}+f_{b}-f_{c}$. Finally, the irregular wake observed for $1.4\leqslant \unicode[STIX]{x1D6FA}\leqslant 3$ is characterized by frequencies $f_{a}$ and $f_{a}-f_{b}$, which display nearly constant values as the rotation parameter increases.

5.1 Qualitative description of flow regimes

The analysis of the transitional wake behind a streamwise rotating sphere is finished with the study of flow regimes at $Re=1000$, which are summarized in figure 16, that displays, for selected values of $\unicode[STIX]{x1D6FA}$, instantaneous plots of vortical structures by means of isocountours of the $Q$-quantity (figure 16a) and time-averaged contours of standard fluctuations of the pressure, $C_{p}^{\prime }$, in the near wake. As in Tomboulides & Orszag (Reference Tomboulides and Orszag2000), an irregular vortex-shedding regime is obtained for $\unicode[STIX]{x1D6FA}=0$, where the dominant frequency is $f_{s}=0.188$. As displayed in figure 16(a), small scales are clearly present at the flow, rendering it turbulent, although large-scale hairpin vortices are evident. As in $Re=500$, there is an azimuthal rotation of the shedding location and, therefore, the hairpin vortices display different orientations as are convected downstream. The shedding intensifies the pressure fluctuations (figure 16b) along the shear layers, as in the irregular regime of $Re=500$.

With rotation, the wake displays similar irregular dynamics, characterized by the rapid rotation and massive separation of vortex loops, of a smaller length scale, which are now distributed along different azimuthal positions. At high values of $\unicode[STIX]{x1D6FA}$, the formation of hairpin vortices seems to take place directly at the shear layer of the sphere. Such modified dynamics strengthen the pressure fluctuations at the rear of the recirculation bubble, as occurs for $Re=500$ and $\unicode[STIX]{x1D6FA}\geqslant 1.4$, suggesting that the wake regimes are similar.

The complexity of the wake dynamics is observable in the phase diagrams of the force coefficients $(c_{y},c_{z})$ and $(c_{x},c_{l})$ of figure 17. The irregular state produces entangled phase portraits, with an increasing magnitude of fluctuations as $\unicode[STIX]{x1D6FA}$ grows. In particular, due to the fostering of the shedding process, the fluctuations of the drag coefficient increase, as shown by the wider diagrams at large $\unicode[STIX]{x1D6FA}$. In addition, a larger radial extension of $(c_{y},c_{z})$ diagrams, which explore statistically the whole azimuth, is seen with the increase of $\unicode[STIX]{x1D6FA}$, indicating an intense wake rotation and a higher eccentricity, that enhances the magnitude of the fluctuating lift coefficient in the $(c_{x},c_{y})$ plot. These portraits are similar to those corresponding to the irregular regime at $Re=500$ and $\unicode[STIX]{x1D6FA}\geqslant 1.4$.

5.2 Frequency analysis

As in the previous sections, we analyse the dominant frequency components from the FFT of fluctuating forces $c_{y}^{\prime }(t)$ and $c_{x}^{\prime }(t)$, and the azimuthal velocity $w^{\prime }(t)$ at the location $(x,y,z)=(2,0.5,0)$, which are represented in figure 18 for selected values of $\unicode[STIX]{x1D6FA}$. As observed, at $\unicode[STIX]{x1D6FA}=0$, the spectra display broad features and it is difficult to infer dominant frequencies from the force coefficients spectra. In both cases, a large energy content exists close to $f=0$, however, in the case of the azimuthal velocity (figure 18c) the shedding frequency can be identified as $f_{s}=0.188$, which, as expected, is slightly higher than that reported for $Re=500$. With the streamwise rotation, the wake displays similar dynamics for the whole interval of $\unicode[STIX]{x1D6FA}$ investigated. In particular, a new frequency $f_{a}$ is discernible at the spectra of $c_{y}^{\prime }$ for $\unicode[STIX]{x1D6FA}\geqslant 0.4$, which is related to the wake rotation, while the corresponding peak associated to the shedding is now modified, corresponding it to $f_{s}+f_{a}$ as observed in the FFT spectra of $w^{\prime }$. In general, the dynamics is characterized by the slow rotation of vortical structures with a frequency $f_{a}$, and the massive shedding of vortex loops, whose frequency $f_{a}+f_{s}$ accelerates with rotation until reaching a nearly constant value at higher $\unicode[STIX]{x1D6FA}$. The spectra are characterized by broader features and multiple frequencies, as expected for the complex and irregular flow behaviour reported earlier. Such seemingly chaotic dynamics render it difficult to identify clear frequency components, along with coherent structures of the dominant modes at the wake, which precluded us from performing such analysis. The complexity and possible chaotic nature of the flow dynamics will be further analysed in § 7.

Figure 17.

Force coefficients at $Re=1000$: (a) phase diagrams of transversal forces ($c_{z},c_{y}$); and (b) phase diagrams of the total lift versus drag ($c_{l},c_{x}$).

Figure 18.

Fast Fourier transform spectra, listing the main frequency components, for different rotation parameters, $\unicode[STIX]{x1D6FA}$, at $Re=1000$, corresponding to the (a) $y$-axis lift coefficient, $c_{y}$, (b) drag coefficient, $c_{x}$, and the (c) azimuthal velocity $w$ at $(x,y,z)=(2,0.5,0)$.