3.1.1. Base flows and pressure by SFD method

In this section we present the base state of the flow past a short rotating cylinder. We focus on the case of ${\small \text{AR}}=1$ and vary the values of $Re$ and $\alpha$. Both temporally mean flow and the steady base flow (solved using the SFD method if unstable) will be displayed and discussed. By extracting pressure field contours and streamlines, we show in figure 2 the typical unstable steady base flow obtained by the SFD method. First, we review the flow past a finite-length cylinder without rotation $(\alpha =0)$ in panels (a,b). For $Re< Re_c=172.2$ (figure 2a), the wake flow is steady, which has two mutually perpendicular symmetry planes $xy$ and $xz$, passing through the geometric centre of the cylinder (as also shown in figure 5 of Yang et al. (Reference Yang, Feng and Zhang2022); the value of $Re_c=172.2$ has also been determined from this work). The wake consists of a recirculation region with closed streamlines, characterized by planar symmetrical separation points and foci. For $Re>Re_c= 172.2$, the flow undergoes a regular bifurcation, resulting in spontaneous symmetry breaking in the $xz$ plane due to nonlinear wake dynamics, as shown in figure 2(b) for the case $Re=330, \alpha =0$. When viewed from the remaining symmetric $xy$ plane with $z=0$ (top panel), one vortex structure in the wake becomes stronger than the other. Besides, the positions of the two separation points slightly change, and the pressure contours indicate that the pressure near the upper part of the cylinder arc surface is larger than that of the lower part, which results in a negative lift coefficient $C_{ly}$. The sign of the $C_{ly}$ in the case of no rotation is unimportant because of the symmetric setting.

Figure 2. Steady SFD base flow past an ${\small \text{AR}}=1$ short cylinder. The colour illustrates pressure contour and the white lines streamlines. Results are shown for (a) $Re=160, \alpha =0$; (b) $Re=330, \alpha =0$; (c) $Re=160, \alpha =0.1$; (d) $Re=330, \alpha =0.1$; (e) $Re=160, \alpha =0.6$; ( f) $Re=330, \alpha =0.6$; (g) $Re=160,$ $\alpha =1.2$; (h) $Re=330, \alpha =1.2$. For each subgraph, plots (a i,b i,c i,d i,e i, f i,g i,h i) show the flow visualisation at the plane $z=0$ and plots (a ii,b ii,c ii,d ii,e ii, f ii,g ii,h ii) at the plane $y=0$. The rotation direction of the cylinder is counterclockwise. The black translucent thick solid lines denote the recirculating region separatrix, which is identified by $U_x=0$.

Figure 2(c,d) show respectively the base flow structures of cases $Re=160$ (stable) and $Re=330$ (unstable) at a low rotation rate $\alpha =0.1$. Even for a low $Re=160$, it can be seen that the rotation breaks the symmetry of the wake, leaving the flow only symmetric with respect to the $oxy$ plane. Compared with the non-rotation case (figure 2a), the positions of the upper and lower separation points shift significantly along the rotation direction. Overall, the asymmetric recirculation region generated by the weak rotation is similar to the asymmetric wake generated by regular bifurcation in the non-rotated case. The counterclockwise rotation makes the flow velocity near the cylinder arc surface on the lower side increase, and the velocity near the upper wall surface decrease. It can be deduced that the pressure near the lower side wall will decrease, and the pressure near the upper side wall will increase, so a negative lift $C_{ly}$ will be obtained, i.e. the Magnus effect. It can be seen from figure 2(c–h) that the numerical results conform to the classical Magnus effect. In the parameter space studied in this paper (especially relatively low $Re$), there is no boundary layer transition from a laminar flow to turbulence, that is, there is no inverse Magnus effect (Kim et al. Reference Kim, Choi, Park and Yoo2014). As mentioned above, the symmetry breaking effect caused by the regular bifurcation resembles that due to the rotation (see also the rotating sphere by Citro et al. Reference Citro, Tchoufag, Fabre, Giannetti and Luchini2016). Therefore, the rotation 'strengthens’ the symmetry breaking caused by the inherent wake mechanism observed in the regular bifurcation without rotation, resulting in the lift force in the rotating cases in figure 2(c–h) being larger, to be shown in figure 4(b). On the other hand, the drag coefficient $C_{d}$ is not very sensitive to the rotation when at low rotation rate $\alpha <0.3$, as shown in figure 4(a).

Figures 2(e) $(Re=160)$ and 2( f) $(Re=330)$ show the base states at moderate rotation $\alpha =0.6$, and both their nonlinear saturation states are steady. So the base flow and mean flow are the same for these cases. Compared with the previous low rotation case, the flow topology further changes, i.e. the recirculation zone in the wake almost disappears, in either the $xy$ or $xz$ plane. But a stagnation point (white point in the panel) similar to that observed in the 2-D rotating cylinder (see figure 3b in Sierra et al. Reference Sierra, Fabre, Citro and Giannetti2020) appears, which is located in the second quadrant of the cylinder.

Continuing to increase the rotation rate, figures 2(g) ($Re=160$) and 2(h) ($Re=330$) show the cases of a high rotation rate ($\alpha =1.2$), and their wake structure and pressure distribution are similar to those of figures 2(e) and 2( f), but the wakes are swung further upwards. The hyperbolic stagnation point (see the white dot) on the upper side moves forward along the direction of tangential velocity, which is similar to the results of Citro et al. (Reference Citro, Tchoufag, Fabre, Giannetti and Luchini2016) on a rotating sphere at a high rotation rate.

3.1.2. Separation bubble

Next, we discuss figure 3 where the streamlines are further analysed. Here, the separation point is defined as the intersection between the envelope of the separation bubble (indicated by the green curve with $U_x(x,y,z=0)=0$ in figure 3a) and the surface of the cylinder, and thereby, the separation angle $\theta _s$ is defined as the angle from the negative $x$ direction to the separation point. In the present rotating cylinder, the values of parameters Reynolds number, rotation rate and aspect ratio have no effect on the separation angle. The separation bubble is always confined between angles $\theta _s=0$ and $\theta _s={\rm \pi}$, as shown in figure 3(a). Note that in the case of a non-rotation cylinder (Yang et al. Reference Yang, Feng and Zhang2022), the separation angles may be varying depending on $Re$ and the aspect ratio. We define the separation bubble length as $x_s=\max \{x-D/2 \mid U_x(x,y,z=0) \leq 0 \}$; see figure 3(a). Figure 3(b) shows $x_s$ as a function of $Re$ for the ${\small \text{AR}}=1$ cylinder, compared with the separation length of a fixed sphere (Johnson & Patel Reference Johnson and Patel1999) and a non-rotating finite-length cylinder (Yang et al. Reference Yang, Feng and Zhang2022). The effect of Reynolds number on $x_s$ is different for the high- and low-rotation-rate cases. When the rotation rate is low, the increase of Reynolds number makes $x_s$ increase monotonically, which is similar to the (mean flow) cases of the fixed finite-length cylinder (Yang et al. Reference Yang, Feng and Zhang2022) at $0.5\leq {\small \text{AR}} \leq 2$ prior to Hopf bifurcation, the sphere (Johnson & Patel Reference Johnson and Patel1999) and rotating sphere (Sierra-Ausín et al. Reference Sierra-Ausín, Lorite-Díez, Jiménez-González, Citro and Fabre2022). The increase of $\alpha$ decreases the value of $x_s$ for $0.025\leq \alpha\leq 1.2$, which means that the rotation shortens the recirculation length. It has been proposed that shortening the recirculation length stabilises the flow; for example, Sierra-Ausín et al. (Reference Sierra-Ausín, Lorite-Díez, Jiménez-González, Citro and Fabre2022) discussed such a stabilising control strategy.

Figure 3. (a) A separation bubble under the effect of rotation in plane $oxy$, whose area is surrounded by the green curve ($U_x(x,y,z=0)=0$) and the wall of the rotating cylinder. The white dot shows the position of the hyperbolic stagnation point. The colour represents the pressure field. (b) Separation bubble length $x_s$ of the SFD base flow as a function of $Re$, compared with a fixed sphere (Johnson & Patel Reference Johnson and Patel1999) and a sphere (Sierra-Ausín et al. Reference Sierra-Ausín, Lorite-Díez, Jiménez-González, Citro and Fabre2022) rotating along the streamwise (blue dashed lines). The blue and black dash-dotted lines represent the $x_s$ obtained from the neutrally stable SFD base flow of a rotating sphere (Sierra-Ausín et al. Reference Sierra-Ausín, Lorite-Díez, Jiménez-González, Citro and Fabre2022) and the present cylinder, respectively. The black and red thick lines represent the $x_s$ obtained from the SFD base flow and the mean flow of a fixed finite cylinder ($\alpha =0$) (Yang et al. Reference Yang, Feng and Zhang2022), respectively.

When the rate $\alpha$ is high, $x_s$ slightly decreases with the increase of $Re$, for example, see $\alpha =0.6,1.2$ in figure 3(b). According to the above discussion on the relation between rotation, recirculation length and flow stability, we cannot make the present rotating cylinder flow more stable at high $\alpha \geq 0.6$-$1.2$ by shortening the base flow recirculation region along the streamwise direction because the value of $x_s$ bounces back when $\alpha \geq 0.6$–$1.2$. We provide a further mechanistic explanation of this result in the structural sensitivity analysis § 3.2.3. Besides, the higher the rotation rate, the less sensitive the value of $x_s$ is to the variation of the Reynolds number.

For fixed $Re$, the effect of rotation rate $\alpha$ on $x_s$ is not monotonous, unlike the sphere rotating along the streamwise axis (Sierra-Ausín et al. Reference Sierra-Ausín, Lorite-Díez, Jiménez-González, Citro and Fabre2022). Sierra-Ausín et al. (Reference Sierra-Ausín, Lorite-Díez, Jiménez-González, Citro and Fabre2022) also suggested that $x_s$ does not have to increase monotonically as $\alpha$ increases after the bifurcation, which can also be seen in the time-averaged recirculation length of the sphere rotating along the streamwise direction (Kim & Choi Reference Kim and Choi2002; Lorite-Díez & Jiménez-González Reference Lorite-Díez and Jiménez-González2020).

3.1.3. Drag and lift coefficients

Now, we turn to figure 4 to discuss the drag and lift coefficients in the short rotating cylinder flow. Starting from the fixed finite-length cylinder case (Yang et al. Reference Yang, Feng and Zhang2022), figure 4(a) shows that the drag coefficient increases as the rotation rate increases, which is caused by the fact that the rotation makes the negative pressure in the leeward area smaller, but has little effect on the maximum positive pressure at the stagnation point on the windward side. These can be seen from the pressure contour in panels (a,b) and (c,d) of figure 2, and more quantitative results on the lift and drag coefficients varying with $\alpha$ are shown in figure 28(a,b) in Appendix C. A comparative analysis reveals that the augmentation of $\alpha$ increases the lift coefficient of both the 2-D infinite cylinder and present 3-D finite-length cylinder. However, in the 3-D case the lift coefficient ceases to increase further when $\alpha$ approaches 2, whereas in the 2-D case the lift coefficient increases monotonically as $\alpha$ approaches 2.5 (Mittal & Kumar Reference Mittal and Kumar2003; Stojković et al. Reference Stojković, Schön, Breuer and Durst2003). The effect of $\alpha$ on drag shows contrasting behaviours between the 2-D infinite cylinder and present 3-D finite-length cylinder cases within $\alpha <2.5$. Note that our analysis is based on the drag and lift in the SFD base flow, whereas the drag and lift for the 2-D infinite cylinder are time averaged (Mittal & Kumar Reference Mittal and Kumar2003; Stojković et al. Reference Stojković, Schön, Breuer and Durst2003).

Figure 4. Drag (a) and lift (b) coefficients of the steady SFD base flow as a function of $Re$ at ${\small \text{AR}}=1$. Comparison with a fixed sphere (Johnson & Patel Reference Johnson and Patel1999), a fixed finite cylinder (Yang et al. Reference Yang, Feng and Zhang2022) and a sphere rotating about the transverse direction (Citro et al. Reference Citro, Tchoufag, Fabre, Giannetti and Luchini2016).

Despite the presence of rotation, it is evident from figure 28(a) that the predominant contributor to the drag is the pressure differential. As the rotational speed increases, the proportion of pressure drag $\bar {C}_{dp}$ to total drag $\bar {C}_{d}$ becomes greater. Similarly, the amplitude of the time-averaged lift coefficient $|\bar {C}_{lp}|$ generated by the pressure differential remains a predominant component of the total lift coefficient $|\bar {C}_{l}|$. Furthermore, emphasis is placed on the primary determinant of drag, namely, the pressure differential. As shown in figure 5, the distribution of spanwise-averaged pressure $P_s$ (panel a) acting on the upper surface, as well as its component $P_{sx}=P_s \cos \theta$ (panel b) along the $x$ axis, is characterized with respect to the angle $\theta$. As an example, the areas corresponding to the shaded regions in panel (b) are denoted as $\varSigma _1,\varSigma _2$ and $\varSigma _3$ for the case $\alpha =2$. So, the drag resulting from the pressure differential can be regarded as the net area enclosed between the blue solid line and the $P_{sx}=0$ axis, specifically $\varSigma _1-\varSigma _2+\varSigma _3$. Consequently, it can be observed that the increase in drag is primarily attributed to a significant reduction (see panel a) in pressure $P_{s}$ on the back surface of the cylinder due to rotation, resulting in a substantial enlargement (see panel b) of the area $\varSigma _3$.

Figure 5. The distribution of span-averaged pressure $P_s$ (a) acting on the upper surface, as well as its component $P_{sx}=P_s \cos \theta$ (b) along the $x$ axis, is characterized with respect to the angle $\theta$. The conditions considered involve cases of ($Re=160, AR=1, \alpha <2)$. As an example, the areas corresponding to the shaded regions in panel (b) are denoted as $\varSigma _1$ (coloured by red), $\varSigma _2$ (coloured by green) and $\varSigma _3$ (coloured by grey) for the case $\alpha =2$.

As with a fixed infinite cylinder (Schlichting & Gersten Reference Schlichting and Gersten2016), a fixed sphere (Johnson & Patel Reference Johnson and Patel1999) and a fixed finite-length cylinder (Yang et al. Reference Yang, Feng and Zhang2022), within the parameter range shown in figure 4(a), an increase in $Re$ generally leads to a decrease in drag coefficient. On the other hand, unlike the drag coefficient $C_d$, the values of the lift coefficient $C_{ly}$ in figure 4(b) present more variation when $Re$ changes. At low rate $\alpha \leq 0.3$, $C_{ly}$ first decreases with the increase of $Re$ and then increases; at high rate $\alpha \geq 0.6$, $C_{ly}$ increases monotonically as $Re$ increases from 50 to 330. Besides, in the interval $0.2<\alpha <0.6$ of figure 4(b), $C_{ly}$ is relatively insensitive to $Re$, and the wake is steady (see the neutral stability curves in the next section). It is worth noting that the lift coefficient $C_{ly} \vert _{\alpha =c}$ of the rotating cylinder and the $C_{ly}\vert _{\alpha =2c}$ of the rotating sphere gradually coincide for $Re\gtrsim 270$ (with $c\leq 0.1$ in present works); see the top-right corner in panel (b). For example, the values of $C_{ly}\vert _{\alpha =0.025}$ of a rotating cylinder and $C_{ly}\vert _{\alpha =0.05}$ of a rotating sphere, or $C_{ly}\vert _{\alpha =0.05}$ of a rotating cylinder and $C_{ly}\vert _{\alpha =0.1}$ of a rotating sphere, are approximately the same for $Re\gtrsim 270$.

3.2.1. At low rotation speed $0\le \alpha \leq 0.3$

We discuss figures 6 and 7 collectively, showing the eigenspectra and the eigenfunctions, respectively. Figure 6 illustrates the influence of parameters ($Re,\alpha$) on the leading eigenmodes at low rotation rate $\alpha \leq 0.3$. The global LSA of the flow past the short rotating cylinder shows that there exist two linear unstable modes (LA and LB) for Reynolds numbers $Re<360$. The linear growth rates $\sigma$ of modes LA and LB increase linearly with $Re$ in the vicinity of the instability, as shown in figure 6(a). When increasing $Re$ for the cases ($\alpha =0, 0.025$ and 0.1), both mode LA and mode LB become unstable through a Hopf bifurcation because the frequency at the neutrally stable condition is non-zero, as shown in figure 6(b). Mode LA undergoes the Hopf bifurcation prior to mode LB at a lower $Re$. At $\alpha =0.15$, the critical $Re$ for mode LB is smaller. From panel b, one can also see that the frequency of global mode LA is not sensitive to $\alpha$ or $Re$ with an approximate value around 0.14, while the frequency of global mode LB increases rapidly with the increase of rotation ratio $\alpha$. Combining the growth rate $\sigma$ and the frequency $St$, we plot in panel 6(c) the modes LA and LB for the representative points LP1–LP5 and the LT point (which can be found in figure 8 for their $Re,\alpha$ values). The implication of panel (c) is similar and, thus, we will not discuss it further.

Figure 6. The growth rate $\sigma$ (a) and frequency (b) of the leading global modes (of the SFD base flow) as a function of $Re$ at ${\small \text{AR}}=1$. The position of the black star symbol ‘’ in panel (a) indicates that $\sigma _{LA}=\sigma _{LB}$. (c) The eigenspectra of global modes LA (grey shaded area) and LB at points LP1, LP2, LT, LP3, LP4 and LP5 in figure 8(a).

Figure 7. Four representative eigenmodes with weak asymmetry in the flow past a short rotating cylinder at $\alpha =0.1$ and $Re=290$. Mode LA (in panel a) and mode LD (in panel d) are similar to mode A and mode B reported by Yang et al. (Reference Yang, Feng and Zhang2022), respectively. The new mode LB (in panel b) with high frequency is more asymmetric caused by rotation. The mode LC (in panel c) has zero frequency. The $Q$-criterion isosurfaces $Q=0$ are coloured by the $x$ component of the vorticity ranging from $-2\times 10^{-3}$ to $2\times 10^{-3}$. The corresponding eigenvalues are (a) LA mode $\lambda _{LA}=4.120\times 10^{-3}+{\rm i}0.8833$, (b) LB mode $\lambda _{LB} = -2.202\times 10^{-2} + {\rm i}1.176$, (c) LC mode $\lambda _{LC} = -6.295\times 10^{-2} + {\rm i}0.0$, (d) LD mode $\lambda _{LD} = -8.683\times 10^{-2} + {\rm i}0.8687$.

Figure 8. (a) Neutral stability curves undergoing the Hopf bifurcation for the flow past a finite rotating cylinder at ${\small \text{AR}}=1$ and the rotating sphere (the mode I by Citro et al. Reference Citro, Tchoufag, Fabre, Giannetti and Luchini2016). The specifications of the points LP1-5 and the codimension-two point LT are shown in table 1. The hollow symbols represent the corresponding nonlinearly saturated wake that looks similarly to global mode LA, whereas the solid symbols denote the nonlinearly saturated wake that more closely resembles global mode LB. (b) Plot of critical Strouhal numbers $St_0$ on the neutral stability curves against rotating ratio $\alpha$.

Figure 7 plots four eigenmodes LA, LB, LC, LD at $\alpha =0.1$ and $Re=290$. Only mode LA is unstable; see also the solid green lines in figure 6(a) and one of the unnamed hollow diamonds on the $Re$-$\alpha$ plane in figure 8(a). The structure of the low-frequency global mode LA is shown in figure 7(a), corresponding to the periodic wake LA (to be discussed in figure 13(a) obtained by DNS) and is mainly caused by the vortices shedding from the flat ends of the cylinder (Yang et al. Reference Yang, Feng and Zhang2022). We loosely use the same name for the global mode and the corresponding wake pattern observed in DNS, if their frequencies are close. Global mode LB, as shown in figure 7(b), with a higher frequency is caused by the vortices shedding from the curved surface and the associated nonlinear flow pattern wake LB is shown in figure 13(b). In figure 7 we have additionally shown two additional modes (named LC and LD) at $\alpha =0.1$ and $Re=290$. Mode LC in panel (c) has zero frequency and presents a long smooth streamwise structure. The leading eigenmode in some other cases, such as LT, may take this form. As shown in figure 8(a), LT represents a low-rotation-rate transition state where the LA and LB modes are both neutral, to be discussed further below. Mode LD in panel (d) appears similar to mode B identified in Yang et al. (Reference Yang, Feng and Zhang2022) for the non-rotating short cylinder. Each global mode in figure 7 maintains the same symmetry as the base flow, namely, symmetry with respect to the $xy$ plane, consistent with that of the non-rotating short cylinder (Yang et al. Reference Yang, Feng and Zhang2022).

Table 1. The specifications of the five typical points in the parameter space $(Re,\alpha )$ (see also figure 8a) for the cylinder ${\small \text{AR}}=1$. Subscripts $_{LA}$ and $_{LB}$ represent global modes LA and LB (figure 7a,b), respectively. The eigenfrequencies marked in bold are close to the frequency $St_{DNS}$ in DNS (e.g., see also figure 14) of the saturated nonlinear wake.

Examining closely the solid and dashed lines in figure 6(a), when $\alpha$ is relatively small, mode LA becomes unstable before mode LB when increasing $Re$. In the range of larger $\alpha$, mode LB starts to become unstable before mode LA. The intersection point of the $\sigma$–$Re$ lines of mode LA and mode LB denotes the condition where $\sigma _{LA}=\sigma _{LB}$, as represented by the black stars in figure 6(a). This competitive phenomenon also exists in spheres rotating along the streamwise direction (Sierra-Ausín et al. Reference Sierra-Ausín, Lorite-Díez, Jiménez-González, Citro and Fabre2022), but differs from the single-mode instability observed in spheres rotating along the transverse direction (Citro et al. Reference Citro, Tchoufag, Fabre, Giannetti and Luchini2016). Note that our cylinder is rotating along its axis that is in the transverse direction.

The neutral curves in the $Re$-$\alpha$ plane associated to the two most unstable modes LA and LB are displayed in figure 8. The present asymmetric steady state is linearly stable in the blank region and linearly unstable in the shaded region, as shown in panel (a). With the rotation speed increasing, the threshold Reynolds number for the instability first decreases slightly and then increases. The critical Reynolds number approximately reads $Re_c=279$ at $\alpha =0.0375$. As the rotational speed continues to increase, the threshold Reynolds number becomes less sensitive to the rotation speed. Thus, the result indicates that the rotation of a short cylinder can influence and control the stability properties of the flow.

The present work designates the overlapping area between the unstable regions of modes LA and LB in figure 8(a) as the bi-unstable region, where points LP2, LP3 and LP4 are located. In this study the intersection of the neutral curves of mode LA and mode LB is denoted as the codimension-two point LT, where two different bifurcations (due to mode LA and mode LB) occur simultaneously. Similar to the rotating sphere (Sierra-Ausín et al. Reference Sierra-Ausín, Lorite-Díez, Jiménez-González, Citro and Fabre2022), the present point LT, as the organizing centre of the linear system, represents a turning point of competition between different modes and results in the generation of three distinct wake patterns around it in the nonlinear system (to be discussed in figure 13). The dotted line in figure 8(a) passes the codimension-two point LT in the bi-unstable region, on which the growth rate $\sigma _{LA}=\sigma _{LB}$. Above this dotted line, the growth rate of mode LB is greater than that of mode LA; vice versa.

The frequencies $St_0$ corresponding to the neutral conditions in panel (a) are reported in figure 8(b) as a function of $\alpha$. As the frequencies are non-zero, the unstable flows will undergo Hopf bifurcation, that is, the unstable mode will oscillate at a certain frequency. Similar to the flow past a sphere rotating along the transverse (Citro et al. Reference Citro, Tchoufag, Fabre, Giannetti and Luchini2016) and streamwise directions (Sierra-Ausín et al. Reference Sierra-Ausín, Lorite-Díez, Jiménez-González, Citro and Fabre2022), the $St_0$ of present rotating cylinders increase rapidly as $\alpha$ increases in the regime of low rotation rates $\alpha <0.3$.

3.2.2. At high rotation speed $0.9<\alpha <2$

We consider relatively low $Re$ in our work. Consequently, in the medium range of the rotation rate $\alpha$ from 0.3 to 0.9 and $Re<360$, the wake is steady without vortex shedding. Thus, we will not discuss this range of parameters.

We continue to study high rotation speeds in the range of $0.9<\alpha <2$ and $Re<360$, where the flow may become linearly unstable as shown in figure 9(a). From this figure, one can observe that the global LSA based on the asymmetric SFD steady state (see figure 2) indicates three unstable modes in this space of parameters, namely HA, HB and HC, whose critical conditions are depicted as three lines in the figure. Four typical pairs of $Re$ and $\alpha$ in this figure, denoted as HP1 to HP4, are highlighted and their HA, HB and HC eigenvalues are quantified in table 2. The four probed points will be further analysed in figure 13. From panel (a) one can see that higher rotation speeds in the range of $0.9<\alpha <2$ can significantly lower the critical Reynolds number for instability, which is different from the low-rotation-rate cases $0\le \alpha \le 0.3$.

Figure 9. (a) Neutral stability curves undergoing the Hopf bifurcation for the flow past a finite rotating cylinder at ${\small \text{AR}}=1$ and the rotating sphere (the mode I by Citro et al. Reference Citro, Tchoufag, Fabre, Giannetti and Luchini2016). The shaded area indicates a region of linear instability. The eigenvalues of points HP1–HP4 are shown in table 2. (b) Plot of critical Strouhal numbers $St_0$ against rotating ratio $\alpha$.

Table 2. The location and eigenvalues of typical points in the parameter space $(Re,\alpha )$ (figure 9a). Subscripts $_{HA}$, $_{HB}$ and $_{HC}$ represent global modes HA, HB and HC (figure 10), respectively. The eigenfrequencies marked in bold are identified in the frequency spectrum (figure 14b–f) of the saturated nonlinear wake.

As shown in panel 9(b), the three eigenmodes HA, HB, HC are characterized by different frequencies $St_0$, namely, low frequency for HA, immediate frequency for HB and high frequency for HC. The non-zero frequencies of these modes again indicate that the unstable wake flow experiences Hopf bifurcation. By comparison to figure 8(b), the influence of the rotation on the eigenfrequencies of HA, HB and HC is less significant compared with the mode LB at low rotation rate. This indicates that in the high-rotation-speed regime, changing the value of $\alpha$ affects less the frequency in the flow. For the following discussions, when a nonlinear wake flow possesses multiple characteristic frequencies of these global modes at the same time, we will name the wake flow by combining the modes; for example, if both the eigenfrequencies of HA and HB modes are observed in a nonlinear wake, we will call it HAB (see figure 13 to be discussed).

Besides, the three unstable modes can also interact with each other, resulting in the three turning points HT1, HT2, HT3 as shown in figure 9(a). In the parameter ranges of $0.9<\alpha <1.64$ and $203< Re<335$, the neutral curves of global modes HA and HB almost coincide, that is, the low-frequency mode HA and mid-frequency mode HB simultaneously become linear unstable. For $\alpha >1.64$ (see, e.g. the point HT1 in figure 9a), the Hopf bifurcation in the flow begins to be dominated by the high-frequency mode HC. In general, the differences of the Hopf bifurcation in our case and that in the rotating sphere are summarised as follows. (1) When $\alpha >0.7$, Citro et al. (Reference Citro, Tchoufag, Fabre, Giannetti and Luchini2016) reports only one unstable mode in the flow past a rotating sphere. However, there are three unstable modes in our case, indicating that the Hopf bifurcation process in the present finite cylinder may be more complex. (2) The eigenfrequency of a sphere is in general higher than that of the rotating cylinder (see the black line with circles in panel 9b).

In figure 10 we show the flow structure of the global eigenfunctions for the three eigenmodes HA, HB, HC at the point HP3 in figure 9. It can be seen that global mode HA (figure 10a) and mode LA (figure 7a) have a similar wake structure, and the difference is that the greater rotation rate makes the transverse offset of mode HA larger. The intermediate-frequency mode HB and the high-frequency mode HC have smaller flow structures than those in mode HA.

Figure 10. The global modes with strong asymmetry for the flow past a rotating finite cylinder at a high rotation rate at $Re=170, \alpha =1.8$ (point HP3). The $Q$-criterion isosurfaces $Q=0$ are coloured by the $x$ component of the vorticity ranging from $-2\times 10^{-3}$ to $2\times 10^{-3}$. The cylinder centroid is located at $(0, 0, 0)$. Results are shown for the (a) HA mode, (b) HB mode, (c) HC mode.

3.2.3. Structure sensitivity

The global modes detailed above characterize the perturbation growth. The sensitivity of the flow to the perturbation can be studied via the structural sensitivity analysis. To further study the destabilization mechanism, such an analysis for the selected low rotation and high rotation flows is conducted. Figure 11 depicts the case of $Re=290, \alpha =0.1, {\small \text{AR}}=1$ for a low rotation case. The transparent red and opaque blue isosurfaces, computed according to $\zeta$ in (2.12), delimit the wavemaker region, which is the superposition of the leading global mode and its adjoint mode. For both the unstable LA and LB modes, we compute their adjoint modes in the global LSA based on the SFD base flow and time-mean base flow, respectively. In general, the wavemaker region is located in the near wake region of the cylinder (in the top-right region of the $xy$ plane). Its structure remains the same symmetry as the base state, being symmetric with respect to the $\mathcal {O}xy$ plane. The spatial distributions of wakemakers based on mean flow and base flow are similar. Since the wavemaker region indicates the most sensitive region in the flow, one can infer from these observations that (i) the region responsible for the instability is located in the recirculation region behind the cylinder, and (ii) the instability mainly amplifies the perturbations near the cylinder surface (Citro et al. Reference Citro, Tchoufag, Fabre, Giannetti and Luchini2016).

Figure 11. Flow sensitivity in a low-rotation-rate case. The wavemaker isosurfaces are plotted for the first two unstable modes LA (a,c) and LB (b,d) at $Re=290,\alpha =0.1,{\small \text{AR}}=1$. Transparent red is for $\zeta =0.2$ and opaque blue is for $\zeta =0.4$. Panels (a,b) show the results of the global LSA based on the SFD base flow and panels (c,d) on the time-mean flow.

Figure 12 presents the flow sensitivity for the high rotation case of HP3, showing the superposition of the global modes HA, HB, HC based on the SFD base flow with their respective adjoint modes. One can see that the three wavemaker regions differ significantly, in contrast to the low rotation case. The wavemaker region for the HB mode is located further downstream of the rotating cylinder compared with the other two modes, whose sensitivity regions are close to the cylinder. This implies that the control of the unstable modes in the high rotation case can be treated separately; especially, control of the HB mode may be achieved more easily as it is not mingled with other modes in the spatial distribution.

Figure 12. Flow sensitivity in a high-rotation-rate case. The most unstable wavemaker isosurfaces are plotted based on the SFD base flow of a rotating cylinder at point HP3. Transparent red is for $\zeta =0.2$ and opaque blue is for $\zeta =0.4$. Panels (a–c) correspond to global modes HA, HB and HC, respectively.

3.3.1. Wakes behind the short rotating cylinder

Some representative spatial structures and the phase diagrams of the lift-drag coefficients are depicted in figure 13 for the short rotating cylindrical wake flow, obtained by the nonlinear DNS. To analyse the wake structure and compare with the results of the global stability analysis, the frequency in the nonlinear saturated system is also computed, i.e. the power spectral density (PSD) in figure 14. The spectra are obtained by calculating the oscillatory part in the time series of the drag coefficient, e.g. $C_d'=C_d-\bar {C}_d$. We calculate the PSD from the drag coefficient, instead of the lift coefficient, because the vortex shedding can take place from the end plates and also the curved surface of the short cylinder, as depicted in figure 1. When the vortices shedding from the end plates (causing spanwise oscillations) are much weaker than those shedded from the curved surface, we noticed that the corresponding oscillation frequencies cannot be observed clearly in the fast Fourier transform (FFT) spectra of the lift coefficient $C_{ly}$. Compared with the lift coefficient, the drag coefficient seems to be a more robust option for analysing the time-history data in our case. When the vortices shed alternately, the frequency of the drag coefficient is twice that of the lift coefficient.

Figure 13. (a i,b i,c i,d i,e i, f i) The nonlinear wakes spatial structure obtained by DNS. The $Q=0$ isosurfaces are coloured by the streamwise vorticity ranging from $-$0.1 to 0.1. (a ii,iii,b ii,iii,c ii,iii,d ii,iii,e ii,iii, f ii,iii) The corresponding phase diagrams of $C_d-C_l$ for a rotating cylinder. (a) Wake LA at point LP2, (b) wake LB at point LP4, (c) wake HA at point HP1, (d) wake HAB at point HP2, (e) wake HC at point HP3 and ( f) chaotic wake HAC at point HP4. The corresponding $Re$, $\alpha$ and eigenvalues for each case are shown in tables 1 and 2.

Figure 14. The PSD for the cases (a) LP2, (b) LP4, (c) HP1, (d) HP2, (e) HP3 and ( f) HP4. The PSD is calculated based on the oscillatory part of the time series of the drag coefficient.

At low rotation, panels 13(a,b) show that the saturated states at points LP2 and LP4 are a limit cycle. Wakes LA and LB represent the wake dominated by vortices shedding from the cylinder’s flat ends and the circular arc surface, respectively. Both types of wake structures, LA and LB, are also observed in the fixed cylinder flow (see wake patterns P3-2 and P3-1 in figure 10 of Yang et al. Reference Yang, Feng and Zhang2022, respectively). The difference is that, due to the rotation effect, wake LB undergoes a Hopf bifurcation and becomes a saturated wake state. In the non-rotating cylinder flow, wake LB is only an intermediate transitional state.

The results of high rotation speeds are shown in panels ($c$–$f$). Three frequencies of oscillations, including low (HA), medium (HB) and high (HC), are identified in the present nonlinear wakes; see previous discussions on figure 9. Wake HA pertaining to the point HP1 (figure 13c) is characterized by a low-frequency oscillation in the spanwise direction and its higher-order harmonics (see figure 14c). Its phase diagram of the lift-drag coefficient is also a limit cycle. Wake HAB (figure 13d) is identified with both a low- and a medium-frequency oscillation at point HP2. The phase diagram indicates a limit torus with two incommensurate frequencies; see also the PSD result in figure 14(d). From the perspective of the flow structure, the medium-frequency oscillation (HB) is caused by the vortices shedding from the cylindrical arc surface, while the low frequency (HA) is associated to the oscillation of the vortices in the $z$ direction. Currently, we are not able to identify a monochromatic wake HB with only a medium frequency. From the vortex street structure of the wake HAB at point HP2, it can be seen that the medium-frequency content exists (referring to the grey-coloured structure in the upper part). In § 3.3.2 we further use the DMD method to decompose the main components in the HAB wake to understand wake HB. Wake HAC (figure 13e) is identified with a high frequency and a low frequency at point HP3, and its phase diagram of the lift-drag coefficient is also a limit torus (see figure 14(e) for the PSD result). Finally, in figure 13( f) a main low-frequency oscillation and its higher harmonics are identified at point HP4, along with a broadband of high-frequency oscillations, leading to a very chaotic signal. Point HP4 is located at the region where the modes HA and HB are both unstable from the linear analysis. The corresponding phase diagram is also more chaotic compared with the previous cases.

3.3.2. Comparison with DMD modes

According to the global LSA results of a non-rotating 3-D finite-length cylinder flow (Yang et al. Reference Yang, Feng and Zhang2022), at the Hopf bifurcation point, we can accurately predict the vortex shedding frequency through the LSA of the time-mean flow. Moreover, the eigenfrequency of the steady base flow (solved using the SFD method if unstable) does not differ too much from the nonlinear vortex shedding frequency. In the present work we want to establish a qualitative/quantitative relationship between nonlinear and linear systems by comparing the frequency and shape of the dynamic mode decomposition (DMD) modes with the linear global modes. The DMD (Rowley et al. Reference Rowley, Mezić, Bagheri, Schlatter and Henningson2009; Schmid Reference Schmid2010, Reference Schmid2022) and its extensions have been applied extensively in flow analyses, and have been tested and proven useful to identify the spatiotemporal patterns of nonlinear flow associated with periodic (Bagheri Reference Bagheri2013, Reference Bagheri2014) and quasiperiodic oscillations (Sierra-Ausín et al. Reference Sierra-Ausín, Lorite-Díez, Jiménez-González, Citro and Fabre2022), transitional regimes (Le Clainche & Vega Reference Le Clainche and Vega2017) and turbulent channel flows (Le Clainche et al. Reference Le Clainche, Izbassarov, Rosti, Brandt and Tammisola2020). The DMD analysis enables a better understanding of the influence of the linear instability on the onset of vortex shedding of the short rotating cylinder flow. For the data to be processed in the DMD analyses, we typically utilised 150 snapshots over five periods of vortex shedding. We made sure that the simulations reached a steady-state vortex shedding state before capturing the data. In cases where the wake exhibited multiple cycles, we considered the longest cycle when determining the period of interest.

We first recall the results in figure 8(a) that below the codimension-two point LT for $\alpha <0.1294$, the steady-state flow transitions supercritically to a wave LA as $Re$ exceeds $Re_c$ (represented by the LA curve in this range of $\alpha$). To demonstrate the supercriticality, we have calculated the Landau coefficient $c_1$ and $c_3$ in Appendix A; see figure 24. Above the codimension-two point, i.e. $\alpha >0.1294$, the steady-state flow transitions to a wave LB with high frequencies. That is, the codimension-two point corresponds to a double Hopf bifurcation, which characterizes the interaction between mode LA and mode LB. In the overlap shaded area shown on the $Re$-$\alpha$ plane in figure 8(a), where modes LA and LB both are linearly unstable in the rotating cylinder flow, we cannot obtain the single-periodic state corresponding to LA or LB separately for low $\alpha <0.3$ by using different initial conditions. That is, regardless of the initial conditions, our nonlinear flow in the low-rotation-rate regime always converges to the most unstable mode. This is in contrast to the results of Sierra-Ausín et al. (Reference Sierra-Ausín, Lorite-Díez, Jiménez-González, Citro and Fabre2022) on a rotating sphere, where they identified a bi-stable region where a single-mode state can be obtained separately and different initial conditions may lead to different flow modes. In figure 15 the DMD spectra of the flows with low rotation rates corresponding to the points LP2 and LP4 in figure 8 are displayed. The results feature fully saturated modes located on the unit circle.

Figure 15. The DMD spectra of point LP2 in panels (a,c) and of point LP4 in panels (c,d). See figure 8(a) and table 1 for the definitions of points LP2 and LP4. Panels (a,c) are on the unit circle and panels (b,d) are on the growth-rate-$St$ plane, where the DMD modes 0, 1, 2 and 3 are marked in red, to be discussed in figure 16.

Figures 16 and 17 show the comparison of the DMD modes with the global modes based on the SFD base flow and time-mean base flow at selected points LP2 and LP4, respectively. Figure 16 shows that DMD mode 0 is the time-averaged flow filed. The DMD mode 1 presents alternated flow structures downstream. According to the imaginary part of the DMD eigenvalues, DMD modes 2 and 3 are the second and third harmonics of mode 1, respectively. At point LP2, the frequencies of the leading DMD mode (panel b) and the leading global eigenmode based on the SFD base flow (panel e) are 0.1436 and 0.1411, respectively, with a difference of $1.7\,\%$. But there is no DMD mode with a frequency close to $\omega =0.1782$, which is also a linearly unstable mode (figure 16f). In the global stability analysis, the (unstable) base flow is solved by the SFD method, whereas our DMD method analyses the saturated flow regime, which hints that the difference in the frequencies in the two methods may be reduced if we use the time-mean flow of the saturated regime in the global stability analysis. This has been carried out and the results are shown in the last row of figure 16 with the subscript $_{MF}$. Now we can see that the frequency of the leading global mode based on the time-mean flow is indeed closer to the leading DMD mode, comparing panel (g) with panel (b).

Figure 16. Comparison of the DMD modes with the global modes at the point LP2 ($\alpha =0.05,Re=305$). Global modes LA (e,g) and LB ( f,h) in the global stability analysis based on the SFD base flow (e, f) and mean flow (g,h). All figures coloured by streamwise vorticity. The eigenfrequencies of the DMD modes are in good agreement with those obtained from the FFT method (figure 14a), with a relative error of $0.48\,\%$. Moreover, the eigenfrequencies of the DMD modes match well with the eigenfrequencies of the mean flow global modes (panel g, f), with a relative error of $0.83\,\%$. However, the difference between the characteristic frequencies of the DMD modes and the eigenfrequencies of the SFD base flow global mode is slightly larger, with a relative error of $2.22\,\%$. Additionally, the topological structure of the DMD modes is almost identical to that of the mean flow global modes. The corresponding eigenvalues are (a) mode 0, $\sigma \,+\,$i$\omega =4.5919\times 10^{-8}\,+\,$i$0.0$; (b) mode 1, $\sigma \,+\,$i$\omega =-1.436\times 10^{-6}\,+\,$i$0.1436$; (c) mode 2, $\sigma \,+\,$i$\omega =-3.5685\times 10^{-5}\,+\,$i$0.2872$; (d) mode 3, $\sigma \,+\,$i$\omega =-6.4764\times 10^{-5}\,+\,$i$0.4309$; (e) ${\rm mode}^{LA}$ $(\sigma \,+\,$i$\omega )_{BF}=3.3957\times 10^{-2}\,+\,$i$0.1411$; ( f) ${\rm mode}^{LB}$, $(\sigma \,+\,$i$\omega )_{BF}=1.6972\times 10^{-2}\,+\,$i$0.1782$; (g) ${\rm mode}^{LA}$, $(\sigma \,+\,$i$\omega )_{MF} =8.4211\times 10^{-3}\,+\,$i$0.1431$ and (h) ${\rm mode}^{LB}$, $(\sigma \,+\,$i$\omega )_{MF}= -4.3530\times 10^{-2}\,+\,$i$0.1915$.

Figure 17. Comparison of the DMD modes with the global modes at the point LP4 ($\alpha =0.1,Re=330$). Global modes LA ( f,h) and LB (e,g) in the global stability analysis based on the SFD base flow (e, f) and mean flow (g,h). All figures coloured by streamwise vorticity. The eigenfrequencies of the DMD modes are in good agreement with those obtained from FFT analysis (figure 14a), with a relative error of $0.05\,\%$. However, the characteristic frequencies of DMD modes are not in complete agreement with those of mean flow and SFD base flow, with errors of $3.03\,\%$ and $4.81\,\%$, respectively. Additionally, the topological structure of DMD modes is nearly identical to that of mean flow global modes, as compared with the SFD base flow. The corresponding eigenvalues are (a) mode 0, $\sigma \,+\,$i$\omega =3.7369\times 10^{-8}\,+\,$i$0.0$; (b) mode 1, $\sigma \,+\,$i$\omega =-5.2874\times 10^{-6}\,+\,$i$0.1976$; (c) mode 2, $\sigma \,+\,$i$\omega = -1.9908\times 10^{-5}\,+\,$i$0.3953$; (d) mode 3, $\sigma \,+\,$i$\omega =-3.8395\times 10^{-5}\,+\,$i$0.5929$; (e) ${\rm mode}^{LB}$, $(\sigma \,+\,$i$\omega )_{BF} =9.3466\times 10^{-2}\,+\,$i$0.2077$; ( f) ${\rm mode}^{LA}$, $(\sigma \,+\,$i$\omega )_{BF}= 4.6550\times 10^{-2}\,+\,$i$0.1451$; (g) ${\rm mode}^{LB}$, $(\sigma \,+\,$i$\omega )_{MF} =5.7977\times 10^{-3}\,+\,$i$0.1917$ and (h) ${\rm mode}^{LA}$, $(\sigma \,+\,$i$\omega )_{MF}= 6.8022\times 10^{-3}\,+\,$i$0.1481$.

Similarly, the DMD and global LSA results for the point LP4 are shown in figure 17. One can see that the frequencies of the leading DMD mode (panel b) and the global mode based on the SFD base flow (panel e) are 0.1976 and 0.2077, respectively, with a difference of $4.9\,\%$. Also, there is no DMD mode with a frequency close to $\omega =0.1451$, which is a linearly unstable mode (figure 17f). The leading global mode based on the time-mean flow in panel g has a closer frequency (0.1917) compared with that based on the SFD base flow to the DMD mode. The global eigenfunction based on the time-mean flow also looks more similar to the DMD mode.

The above results pertain to the low-rotation-rate cases. Next, we compare the linear and nonlinear results for the high-rotation-rate flows. We plot the DMD eigenspectra in figure 18 for the selected HP2 and HP3 cases. We discuss the DMD modes labelled in red in the figures. For HP2, the DMD modes are plotted in figure 19(a,b) along with the global modes based on the SFD base flow (c,d) and the time-mean flow (e, f). Compared with the case of a low rotation rate in figures 16 and 17, the similarity of the DMD modes with the global modes in the high-rotation-rate case is greater; for example, the first column in figure 19 shows that the three modes look similar, and the global mode based on the mean flow is again slightly better compared with the DMD mode. In the second column, we can also find DMD mode 2 that resembles the global modes in the global stability analyses, where the global mode based on the time-mean base flow looks closer to the DMD mode. The same conclusion can be drawn for the HP3 point in figure 20. As the discussions are similar, we will not go into detail about them. To sum up, through the comparison of the frequencies and the shapes of the linear global modes and DMD modes, we can establish a connection between the linear and nonlinear systems.

Figure 18. The DMD eigenvalues spectrum of points HP2 (a,b) and HP3 (c,d) on the unit circle and on the growth-rate-$St$ plane.

Figure 19. Comparison of the DMD modes (a,b) with the global SFD base flow modes (c,d) and mean flow modes (e, f) at point HP2. All figures coloured by streamwise vorticity. The eigenfrequencies of the DMD modes are in good agreement with those obtained from FFT analysis (figure 14d), with a relative error of $0.05\,\%$. Because the point HP2 is very close to the neutral curve, the eigenvalues of both mean flow and SFD base flow can predict the true frequency well, with errors of $0.2\,\%$ and $0.8\,\%$, respectively. The corresponding eigenvalues are (a) mode 1, $\sigma \,+\,$i$\omega =-9.660\times 10^{-7}\,+\,$i$0.03994$; (b) mode 2, $\sigma \,+\,$i$\omega = 7.360\times 10^{-6}\,+\,$i$0.1963$; (c) ${\rm mode}^{HA}$, $(\sigma \,+\,$i$\omega )_{BF} =-7.499\times 10^{-5}\,+\,$i$0.04026$; (d) ${\rm mode}^{HB}$, $(\sigma \,+\,$i$\omega )_{BF}= 1.905\times 10^{-3}\,+\,$i$0.1968$; (e) ${\rm mode}^{HA}$, $(\sigma \,+\,$i$\omega )_{MF} =7.740\times 10^{-4}\,+\,$i$0.04004$ and ( f) ${\rm mode}^{HB}$, $(\sigma \,+\,$i$\omega )_{MF}= -3.019\times 10^{-3}\,+\,$i$0.1964$.

Figure 20. Comparison of the DMD modes (a,b) with the global SFD base flow modes (c,d) and mean flow modes (e, f) at point HP3. All figures coloured by streamwise vorticity. The eigenfrequencies of the DMD modes are in good agreement with those obtained from FFT analysis (figure 14e), with a relative error of $0.7\,\%$. However, the frequencies of DMD modes are not in complete agreement with those of mean flow and SFD base flow, with errors of $5.8\,\%$ and $6.6\,\%$, respectively. Additionally, the topological structure of DMD modes is nearly identical to that of mean flow global modes, as compared with the SFD base flow. The corresponding eigenvalues are (a) mode 1, $\sigma \,+\,$i$\omega =3.307\times 10^{-6}\,+\,$i$0.06350$; (b) mode 2, $\sigma \,+\,$i$\omega = -1.165\times 10^{-6}\,+\,$i$0.2686$; (c) ${\rm mode}^{HA}$, $(\sigma \,+\,$i$\omega )_{BF} =-2.261\times 10^{-2}\,+\,$i$0.06755$; (d) ${\rm mode}^{HC}$, $(\sigma \,+\,$i$\omega )_{BF}= 3.050\times 10^{-2}\,+\,$i$0.2646$; (e) ${\rm mode}^{HA}$, $(\sigma \,+\,$i$\omega )_{MF} =-5.764\times 10^{-2}\,+\,$i$0.06698$ and ( f) ${\rm mode}^{HC}$, $(\sigma \,+\,$i$\omega )_{MF}= 1.673\times 10^{-2}\,+\,$i$0.2707$.