Wake modes behind a streamwisely oscillating cylinder at constant and ramping frequencies
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Abstract
In this paper, the wake modes behind a circular cylinder under streamwisely forcing oscillating motion are studied at Reynolds number Re = 360–460 which are observed by laser-induced fluorescence flow visualization technique. The forcing frequency \(f_{\mathrm{e}}\) ranges from 0 to 6.85 \(f_{\mathrm{s}}\) , where \(f_{\mathrm{s}}\) is the vortex shedding frequency behind a stationary cylinder, and the forcing amplitude \(A/d = 0.2, 0.5, 1.0\), where d is the cylinder diameter. Both time-invariant and linearly ramping \(f_{\mathrm{e}}\) are investigated. Following our previous modal notation, the following conclusions can be drawn: firstly, three rarely reported modes in numerical studies, C-I, C-II and S-III, are now confirmed in experiments (though with differentiable appearance in their far wake behaviour) at higher A and/or \(f_{\mathrm{e}}\) ranges and the envelope lines of S modes and C modes, yielded from a vortex circulation model, are shown to be dependent on the peak relative velocity of the free stream to the cylinder surface. So is the occurrence of the S-II mode. Secondly, near the demarcation of A modes and S-I mode, wake mode undergoes constant transition in a stochastic manner at fixed Re, A and \(f_{\mathrm{e}}\). Thirdly, a typical hysteretic effect can be observed when the oscillation frequency of the cylinder ramps up and down in a linear way, and the extent of delay is dependent on the ramping rate k. Finally, mode switching during frequency ramping obeys a unidirectional order. During \(k<0\) (ramp-down), when S-I (Type-II) mode switches to A-IV mode, or A-IV to A-III, the flow structure downstream is affected by the upstream and the entire wake flow eventually switched, which is classified as slow switches. In contrast, during \(k>0\) (ramp-up), a clear and abrupt switch can be observed in the wake when A-IV or A-III switch to S-I (Type-II) modes, which are jump switches.
Graphical abstract
Keywords
Streamwise oscillation Vortex shedding WakeList of symbols
- A
Forcing amplitude
- l
Spanwise length of the cylinder
- d
Cylinder diameter
- \(U_0\)
Free stream velocity
- \(\nu \)
Kinematic viscosity of the working fluid
- Re
Reynolds number based on the free stream velocity \(Re = U_0d/\nu \).
- \(f_{\mathrm{e}}\)
Fixed forcing frequency of cylinder
- t
Time
- k
Ramping rate
- \(\phi _0\)
Arbitrary starting phase
- \(v_{\mathrm{c}}\)
Linear velocity of the oscillating cylinder \(v_{\mathrm{c}} = 2\pi Af_{\mathrm{e}}\sin (2\pi f_{\mathrm{e}}t+\phi _0)\)
- \(f_{\mathrm{s}}\)
Vortex shedding frequency behind a stationary cylinder
- St
Strouhal number \(St = f_{\mathrm{s}}U_0/d\)
- X(t)
Motion displacement \(X(t) = A\cos (2\pi f_{\mathrm{e}}t+\phi _0)\)
- x
Streamwise direction
- y
Transverse direction
- \(\varGamma \)
Circulation of individual vortex packet
- f(C-I)
Frequency of vortex shedding of C-I mode at \(x\approx 4d\)
- \(Re_{\mathrm{c}}\)
Critical Reynolds number for the formation of S-II mode
- \(V_{\mathrm{i}}\)
Induced velocity
- \(Re_{\mathrm{p}}\)
Peak (relative) Reynolds number \(Re_{\mathrm{p}} = 2\pi f_{\mathrm{e}}AD/\nu +Re\)
- \(\varDelta Re\)
Relative Reynolds number \(\varDelta Re = (2\pi f_{\mathrm{e}} A-U_0)D/\nu \)
- \(\varGamma _{\mathrm{cp}}\)
Circulation of the vortices formed by the \(+\,x\) relative velocity
- \(Re_{\mathrm{cp}}\)
Circulation-based Reynolds number: \(Re_{\mathrm{cp}} = \varGamma _{\mathrm{cp}}/\nu \)
- N
Number of oscillation cycles spent in change from one frequency ramping to another
- \(N_{\mathrm{rp}}\)
Number of cycles of the entire ramp-up/down period.
1 Introduction
The wake flow behind a circular cylinder under sinusoidal oscillation motion in streamwise direction is of both fundamental and practical relevance. The oscillation motion relative to the incoming flow can be generated not only by the motion on the cylinder itself, but also by an unsteady periodic perturbation superimposed on the uniform free stream and impact on a stationary cylinder. Oil or gas platform riser cables and the bridge piers are sample cylindrical-shaped structures exposed to ocean currents or winds, which are often highly unsteady in the form of waves of various wavelengths. Compared with classical Karman vortex street behind a stationary cylinder in a steady uniform in coming flow, if vibrations induced by the periodic incoming flow occur on flexibly mounted cylinders, which can be in both streamwise and transverse direction, the vortex shedding behaviour will become significantly complex, due to the combination of the unsteady motions on both the incoming flow and the structure. It may cause drag and lift on the structure to change rapidly and also those on the downstream one, which further induces fluttering or galloping and hence structural fatigue damage. While wake behind a cylinder under transverse vibration motion, either induced or forced, has been studied extensively (e.g. Williamson and Govardhan 2004; Williamson and Roshko 1988), wake pattern behind a cylinder under streamwise oscillation motion has attracted less attention, which, however, is equally important to problems like flow-induced vibration and flow control (Naudascher 1987; Sarpkaya 2004).
If the incoming flow is uniform and steady, at the right condition, transverse oscillation will occur spontaneously on the cylinder, induced by the alternative vortex shedding, while streamwise oscillation usually requires external forcing to result in a reasonable amplitude, i.e. according to Konstantinidis (2014) and Konstantinidis and Bouris (2016), the amplitude of a spontaneous streamwise oscillation is typically smaller than 0.15d, where d is the cylinder diameter. In such a flow condition, in addition to Reynolds number, \(Re = U_0 d/\nu \), where \(U_0\) is the free stream velocity and \(\nu \) is the kinematic viscosity of the working fluid, the flow is further controlled by the frequency of the external oscillatory driving force, \(f_{\mathrm{e}}\) and the amplitude A. In dimensionless forms, the two governing parameters are A / d and the frequency ratio \(f_{\mathrm{e}}/f_{\mathrm{s}}\), where \(f_{\mathrm{s}}\) is the vortex shedding frequency from a stationary cylinder under the same Re.
Tanida et al. (1973) measured lift and drag forces on a streamwisely oscillating circular cylinder to study the stability of the oscillation motion at \(A/d = 0.14\) and \(f_{\mathrm{e}}/f_{\mathrm{s}} = \) 0–2.0. They found that the streamwise oscillation causes the so-called synchronization in a range around double the Strouhal frequency, viz \(f_{\mathrm{e}}/f_{\mathrm{s}} =\) 0.77–1.54 (\(Re=80\)). They thought that the fluctuating forces consist of two parts: one is due to the vortex shedding, and the other is resulted from the cylinder oscillation. Griffin and Ramberg (1976) visualized the vortex formation from a circular cylinder oscillating in line with the incoming flow at \(Re = 190\). They found that the vortex shedding is all in the ‘lock-on’ condition, where the vortex shedding frequency coincides with that of the structural oscillation frequency and near twice the Strouhal frequency, i.e. in the range of \(f_{\mathrm{e}}/f_{\mathrm{s}} =\) 1.74–2.2 and \(A/d =\) 0.06–0.12. Two distinct wake patterns (A-I and A-III) are also reported for the first time. Ongoren and Rockwell (1988a, b) investigated the wake pattern when \(A/d = 0.13\) and 0.3, \(0.5<f_{\mathrm{e}}/f_{\mathrm{s}}<4.0\). They identified two basic modes, which are the symmetric and anti-symmetric vortex formation, and further classified these two basic modes to five submodes: S mode for the symmetric vortex formation and A-I, II, III, IV modes for the anti-symmetric vortex formation. Cetiner and Rockwell (2001) studied the lock-on state of a streamwisely oscillating circular cylinder in a cross flow (\(0.3<f_{\mathrm{e}}/f_{\mathrm{s}}<3.0\)) and found that the time-dependent transverse force was phase-locked to the circular cylinder motion and the vortex system appeared at both upstream and downstream of the cylinder.
Detemple-Laake and Eckelmann (1989) carried out smoke-wire flow visualization of the wake patterns in a sinusoidally sound superimposed open-circuit wind tunnel at \(Re =\) 60–200. Three types of lock-on modes of wake patterns, with rich details in both streamwise and spanwise directions, were presented and discussed with limit cycles. Nishihara et al. (2005) measured the fluid forces acting on a cylinder forced to oscillate in the streamwise direction and showed the corresponding detailed flow visualization of the wake patterns at \(A/d = 0.05\) for a range of reduced velocities in a water tunnel at subcritical Reynolds numbers. They illustrated the relationship between the damping coefficients and the wake patterns.
Xu (2003) and Xu et al. (2006) increased the oscillation amplitude from \(A/d =\) 0.5–0.67 and found the S-II mode, which consists of two rows of binary vortices symmetrically arranged about the wake centreline. By decomposing the vorticity production into two components that associated with the oscillation of a cylinder in quiescent fluid and that associated with the flow past a stationary cylinder, they concluded that the critical A / d at which the S-II mode occurs scales with \((f_{\mathrm{e}}/f_{\mathrm{s}})^{-1}\). Konstantinidis and Balabani (2007) found that S-II mode could rapidly break down and give rise to an anti-symmetric arrangement of vortex structures further downstream. The downstream wake may or may not be phase-locked to the imposed oscillation.
In addition to the experiments, some numerical simulations have also been applied to investigate similar problems, which enriched dynamic data for the wake mode structures. For example, Song and Song (2003) suggested that the primary and the secondary vortices of S mode are generated by the instability of the vortex sheet and the forcing motion on the cylinder, respectively. Yufei et al. (2007) performed a detailed investigation in lock-on region and proposed \(Af_{\mathrm{e}}^2/df_{\mathrm{s}}^2\) to be an important control parameter for different vortex shedding modes, which successfully predicts the symmetric S mode. Leontini et al. (2013) examined the impact on the vortex shedding frequency of A and \(f_{\mathrm{e}}\) of the oscillation, as well as Re of the incoming flow. The observed declined rates of the frequency with respect to A are shown to be able to predict the oscillation amplitude A on the cylinder when synchronization occurs. Zhou and Graham (2000) computationally studied cylinder placed in oscillatory currents and observed vortex pattern in the wake similar to the experimental visualization by Couder and Basdevant (1986). However, differences in the far-field dissipation of vortex were noticed. Sarpkaya et al. (1992) carried out numerical simulations in so-called laminar pulsatile flows and revealed that wake comprises of three rows of heterostrophic vortices certain oscillation conditions.
Summarizing these previous studies, in which the maximum oscillation amplitude \((A/d)_{\mathrm{max}}<0.8\) and the maximum forcing frequency \((f_{\mathrm{e}}/f_{\mathrm{s}})_{\mathrm{max}}<3\), six wake modes have been identified behind a streamwisely oscillating cylinder, namely the anti-symmetric A-I, II, III, IV modes and the symmetric S-I, II modes. Moreover, the above-mentioned works were conducted under time-invariant \(f_{\mathrm{e}}\) exclusively. The present work aims to explore possible new wake modes at a higher A / d and \(f_{\mathrm{e}}/f_{\mathrm{s}}\) range and provide empirical model for their envelope lines. Moreover, the transient mode switching under the effect of continuous time-variant forcing frequencies, which is more likely to occur in the real world unsteady flow conditions, is investigated for the first time, to the best of the authors’ knowledge.
2 Experimental details
The measurements are performed on the central section of the test cylinder, so as to minimize three-dimensional flow effects. Rhodamine Dye (6G 99%), which turns metallic green colour when excited by a laser of 532 nm wavelength, was introduced at the mid-span through two injection pinholes located at \(\pm \,90^{\circ }\) on the cylinder surface (the leeward stagnation point being \(0^{\circ }\)), as shown in Fig. 1. With a valve controlling the flow rate, dye came out from the pinholes by the hydraulic head created by a dye reservoir, which was placed about 0.8 m above the free water surface in the channel. The head was carefully adjusted to compromise the injection momentum, which is negligible compared to the free stream momentum but is enough to create a flow rate which sufficiently contrasts the subtle wake structures from the background flow. The size of the pinholes is about 0.15 mm in diameter. The near-field wake region was illuminated by a thin sheet of about 2 mm thick emitted from a 10 W continuous wave laser. The field of view (FOV) is about \(-1\le x/d\le 15\) and \(-4\le y/d \le 4\) in the streamwise and spanwise directions, respectively, where \((x,y) = (0,0)\) is at the cylinder centre. A professional digital video camcorder (SONY PXW-X280) was used to record the wake flow at 25 frames per second (fps). Measurements were conducted at Re = 360–460.
3 Wake modes induced by constant \(f_{\mathrm{e}}\)
In the lock-on regime, the wake mode behind a streamwisely oscillating cylinder depends on the combination of A / d and \(f_{\mathrm{e}}/f_{\mathrm{s}}\). The present LIF visualization confirms the occurrence of five basic modes reported previously, albeit at different A / d, \(f_{\mathrm{e}}/f_{\mathrm{s}}\) combinations. The conditions for the occurrence of these modes in the present study are given in Sect. 3.3.
We also observed three modes which have not been discussed well previously, to the best of our knowledge. They typically occur at high A / d and/or \(f_{\mathrm{e}}\)/\(f_{\mathrm{s}}\) ranges, which are denoted as S-III, C-I and C-II modes. These modes are schematically described in Fig. 2, together with the terminology of other modes. It is worth mentioning that S-III mode is similar to the vortex pattern observed by Sarpkaya et al. (1992) numerically. Nevertheless, the formation process was not discussed in detail and the far-field wake seems to have higher viscous effect, even at higher Re, which might be owing to the forced two-dimensional condition in the simulation (while three-dimensional effect is inevitable in experiments), especially in the far field. Moreover, in our experiments, we also observe two submodes of S-III, and their formation conditions will be briefly discussed next.
3.1 C-I and C-II modes
Frequency of the reorganized C-I mode at \(A/d = 0.2\)
Re | \(f_{\mathrm{e}}/f_{\mathrm{s}}\) | f(C-I)/\(f_{\mathrm{s}}\) | f(C-I)/\(f_{\mathrm{e}}\) |
---|---|---|---|
360 | 5.29 | 0.787 | 0.149 |
360 | 6.18 | 0.787 | 0.128 |
360 | 6.85 | 0.91 | 0.132 |
430 | 4.98 | 0.85 | 0.171 |
430 | 5.97 | 0.90 | 0.151 |
460 | 4.60 | 0.93 | 0.201 |
3.2 S-III mode
However, when the strength and the duration of the relative motion of the flow to the cylinder surface (in \(-\,x\) direction) are large enough, the shear layer induced by such a relative motion is not able to fully entrained into the vortex pair (C, D) for it to continue growing; instead, (C,D) will move along with (A, B) under the influence of the latter and the residual shear layer will roll up to a new vortex pair (E, F) near \(t = t_3\). By now, three symmetric pairs of vortices appear in the wake, hence the name S-III.
At some point between \(t_3\) and \(t_4\), \(U_0\) starts to overtake \(v_{\mathrm{c}}\). This happens when \(2\pi f_{\mathrm{e}} A \sin (2\pi f_{\mathrm{e}} t+\phi _0) = U_0\), viz. \(2\pi f_{\mathrm{e}} t+\phi _0 \approx 0.675\). Consequently, (A, B) and (C, D) convect away from the cylinder in \(-\,x\) direction and (E, F) moves to the leeward side of the cylinder while \(\varGamma _{E,F}\) increasing. For \(t>t_4\), the cylinder starts a new cycle, and a new pair (G, H) forms in the same way as (A, B). Induced by the pair (G, H), vortexes E and F move towards each other, which explains the formation process of the vortex row along the wake centreline.
The pairs (A, B) and (C, D) shown in Fig. 8 tend to move away from the wake centreline as they are convected downstream. Their trajectory is mainly determined by the relative magnitude of \(\varGamma _{A,B}\) and \(\varGamma _{C,D}\) and their mutual induced velocities. In this case, \(\varGamma _{A,B} \ge \varGamma _{C,D}\) (conjectured from the dye pattern). However, at the same A and \(f_{\mathrm{e}}\), when \(U_0\) (Re) increases, \(\varGamma _{A,B}\) increases accordingly (since vortices A and B are formed when the cylinder moves upstream, the relative velocity increases), so for \(\varGamma _{G,H}\). But \(\varGamma _{C,D}\) decreases, which results in \(\varGamma _{A,B}>\varGamma _{C,D}\). Considering the induced velocity, \(\varGamma _{C,D}\) and \(\varGamma _{E,F}\) tend to pull (A,B) away from the centreline, while \(\varGamma _{G,H}\) brings them inward. Since \(\varGamma _{G,H}\gg \varGamma _{C,D}, (\varGamma _{E,F})\), (A, B) quickly entrains (C, D) and moves towards the centreline together, as shown in Fig. 9. They then merge with (E, F) to form a symmetric vortex pair of larger size at the centreline.
As expected, similar to submodes Type-I and Type-II of S-III, which are dependent on the motion of the outer pairs (A, B) and (C, D), two sub types can also be observed in S-II mode, according to the same mechanism. This is shown in Appendix 1. As the forcing frequency goes up, wake mode gradually transfers from Type-I to Type-II. Compared to Figs. 8 and 9, in the absence of (E, F), the vortex pair at the wake centreline in S-III Type-II is less distorted, but it also gets unstable and disorganized soon moving downstream.
3.3 The mode map
If the oscillation energy increases to \(\varGamma _{\mathrm{cp}}/\nu \ge 70\) while \(Re_{\mathrm{p}} \le 1300\), the relative flow motion to the cylinder in \(+\,x\) direction will roll up to two +ve vortices and S-III mode form, as described in Sect. 3.2.
Thirdly, when \(\varDelta Re < 0\), \(2 \pi f_{\mathrm{e}}A < U_0\), the free stream flow will always be in \(+\,x\) direction with respect to the cylinder and S-II/S-III mode will not occur. Only +ve vortices will form as S-I mode or A modes in the lock-on regime. An interesting observation is that at A / d = 0.2 for \(1< f_{\mathrm{e}}/f_{\mathrm{s}} < 4\) , S-I modes are separated by a range of A modes. The wake patterns of the S-I mode at \(f_{\mathrm{e}}/f_{\mathrm{s}} \approx 1\) and that at \(f_{\mathrm{e}}/f_{\mathrm{s}} >2\) (for all three Re) appear differently and are denoted as Type-I and Type-II, respectively. These are discussed in Appendix 2.
In addition, not all of the modes can be observed at a fixed oscillation amplitude A. The higher the A is, the more modes which will be skipped. For instance, at \(A/d = 0.2\), most of the modes can be observed, while at A / d = 1.0, mode A-I \(\rightarrow \) S-III are all skipped. The appearance of mode A-I is very subtle, which can only be seen at an extremely narrow frequency band at Re = 360, A / d = 0.2, \(f_{\mathrm{e}}/f_{\mathrm{s}} \approx 1\). The mode distribution is found to be a weak function of Re, which can also be seen in Fig. 11. However, as Re increases, the frequency band for both non-lock-on and S-I/S-II/S-III modes shrinks, which agrees with Xu et al. (2006).
Figure 14 presents three instants where mode transition occurs at a fixed forcing condition, which illustrates the transition mechanism. Although Fig. 14 only shows one-way transitions, in experiments, we observed that transitions in the reverse order also take place and equally frequently. It is worth mentioning that in Ongoren and Rockwell (1988b), at A / d = 0.2, \(f_{\mathrm{e}}/f_{\mathrm{s}}\)(A-III) \(< f_{\mathrm{e}}/f_{\mathrm{s}}\)(A-IV), whereas Fig. 11 shows that \(f_{\mathrm{e}}/f_{\mathrm{s}}\)(A-III) \(> f_{\mathrm{e}}/f_{\mathrm{s}}\)(A-IV). This particular discrepancy is believed to be partly due to different visualization techniques, which influence the rather subjective judgement, and partly due to the unstable nature of the two modes, where transition between them can be caused by very small instability.
4 Wake modes induced by linear ramping \(f_{\mathrm{e}}\)
4.1 Mode switches
4.2 Mode skip and hysteresis
Secondly, fewer modes are identified when the forcing frequency changes quickly, i.e. as |k| increases, some modes are skipped. For instance in Fig. 15, as k increases from + 0.04\(s^{-2}\) to + 0.08\(s^{-2}\), mode A-III is skipped while A-IV can still be observed, even though its frequency band becomes narrower, which suggests that mode A-IV is more stable. As k is set at the maximum ramping rate + 0.44\(s^{-2}\), only one lock-on mode A-IV can be seen. Similar observations can be made for \(k < 0\) cases. The number of modes that can be identified dropped from 4 at \(k = -\,0.02s^{-2}\) to 1 (only S-II) at \(k = -0.04 s^{-2}\). Note that at k = ± 0.44 \(s^{-2}\), the total number of oscillation cycles spent in the ramping period is only N = 4 according to Eq. 8, which leaves no time for more modes to emerge.
Thirdly, during the ramp-up period, the occurrence of S-I (Type-II) and S-II modes is clearly delayed in terms of the occurrence \(f_{\mathrm{e}}\) compared to the base modes, as indicated by the horizontal grid lines. Such a delay becomes more significant as k increases. Similarly, the corresponding \(f_{\mathrm{e}}\) for the occurrence of mode A-III (if possible), A-IV and S-I (Type-II) or S-II also gets delayed, while the frequency bands for the non-lock-on mode and no clear mode expand. During the ramp-down period, on the other hand, the terminal \(f_{\mathrm{e}}\) for S-I (Type-II) mode range is postponed compared to the base modes, especially as |k| increases. Also, the corresponding \(f_{\mathrm{e}}\) for the occurrence of slow switch and A-IV mode is also gradually deferred, with the frequency band of slow switch enlarged marginally.
Consequently, the most striking feature of Fig. 15 perhaps is the asymmetry of the modal distribution during the ramp-up and ramp-down periods. Not only the modes themselves, but also their occurrence \(f_{\mathrm{e}}\) exhibit asymmetry. Although the mode divisions in Fig. 15 along a constant k line are approximations, the hysteretic effect is discernable, as shown by the marked horizontal grid lines. As discussed above, hysteresis also exists between the ramp-up and the base modes. Figure 22 presents an example of hysteresis for mode switches from A-IV to S-I (Type-II).
Hysteresis \(\varDelta f_{\mathrm{e}}\) generally increases for other modes at higher k, which can be inferred by Fig. 15. At the largest \(|k|= 0.44 s^{-2}\), mode S-I (Type-II) is not seen in the ramp-up period until the k = 0 at the plateau, while mode switch from S-I (Type-II) does not occur in the ramp-down period. Hysteresis can thus be considered as at maximum degree, since \(\varDelta f_{\mathrm{e}}\) is not defined. Figure 23 presents the large-scale symmetric structure appears when \(f_{\mathrm{e}}\) goes through linear ramp-up or down rapidly.
5 Conclusion
In this paper, the wake modes behind a circular cylinder in streamwisely oscillating motion are studied at higher forcing frequencies under Re = 360–460, using LIF flow visualization. The forcing frequency \(f_{\mathrm{e}}/f_{\mathrm{s}}\) ranges from 0 to 6.85 and the amplitude A / d(= 0.2,0.5,1.0). Both time-invariant and linear ramping \(f_{\mathrm{e}}\) are investigated. The following conclusions can be drawn:
(1) Three rarely reported modes (C-I, C-II and S-III) are identified at higher A and/or \(f_{\mathrm{e}}\) ranges than those applied in previous studies. The appearance of the lock-on modes at a fixed A generally follows the order S-I (Type-I)\(\rightarrow \)A modes\(\rightarrow \)S-I (Type-II)\(\rightarrow \)S-II/S-III\(\rightarrow \)C modes, as the time-invariant \(f_{\mathrm{e}}\) increases. When \(f_{\mathrm{e}}\) is close to the demarcation frequency between A modes and S-I(Type-II) mode, i.e. at A / d = 0.2 and \(2.14 \le f_{\mathrm{e}}/f_{\mathrm{s}} \le 2.24\) depending on Re, unsteady bidirectional mode transitions are observed, where multiple modes coexist and transfer back and forth from one to another. At fixed \(f_{\mathrm{e}}\), the additional constraint for the occurrence of S-II, C-I and C-II modes are also modelled from which, envelope lines for these modes are marked on the mode map based on empirical parameters.
(2) More new observations can be made as the oscillation frequency \(f_{\mathrm{e}}\) undergoes linear ramping. As the magnitude of the ramping rate |k| increases, more modes are skipped compared to the base modes (k = 0) for the same \(f_{\mathrm{e}}\) range. Also, the mode distribution is found to be asymmetric between ramp-up (\(k>0\)) and ramp-down (\(k<0\)) ranges, which is a typical hysteretic effect. By and large, \(f_{\mathrm{e}}\) for a certain mode to occur is deferred in \(k>0\) compared to k = 0 cases, while it is further postponed during the \(k<0\) range. The degree of delay is in line with |k|, i.e. the larger the ramping rate, the larger the hysteresis effect. At the largest |k|= 0.44\(s^{-2}\), which is the highest ramping rate tested, except the base mode S-II, only A-IV appears for a small \(f_{\mathrm{e}}\) range during ramp-up, while no other modes can be seen during ramp-down.
(3) Mode changes during frequency ramping all obey unidirectional switches. However, the switch processes are different for ramp-up and ramp-down durations. During \(k < 0\), when S-II or S-I (Type-II) mode switch to A-IV mode, or A-IV to A-III, the flow structure in downstream is affected by the upstream and the entire wake flow eventually switched, which is a slow switch. In contrast, during \(k > 0\), a clear and abrupt switch can be observed in the wake when A-IV or A-III switch to S-I (Type-II) modes. This type of switch is named jump switch.
Notes
Acknowledgements
S.J. Xu wishes to acknowledge the support from NSFC through Grants 11772173, 91752203 and 11472158. L. Gan would like to thank the support from Durham University International Engagement Grant.
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