4.1.1. Undeformed equilibrium: state I

For very low $ {\kappa }$, the rigid inverted flag remains completely stationary. The flow is dominated by a wake consisting of two long shear layers produced from the boundary layers on each side of the flag. Figure 5 shows an example of this flow and the associated time history of the angular displacement of the flag, $\theta$, which is zero throughout.

4.1.2. Deformed equilibrium: state II

As $ {\kappa }$ is increased, the first bifurcation occurs with the rigid inverted flag deflecting steadily to one side. The wake flow structure is almost unaffected relative to state I, except for a slight asymmetry; see figure 5. A similar biased state has been observed for the flexible inverted flag at a similar Re (Gurugubelli & Jaiman Reference Gurugubelli and Jaiman2015; Ryu et al. Reference Ryu, Park, Kim and Sung2015; Goza et al. Reference Goza, Colonius and Sader2018).

The rigid inverted flag acts as a thin airfoil that is prone to divergence instability; similar to a flexible inverted flag (Sader et al. Reference Sader, Huertas-Cerdeira and Gharib2016b). We consider a generalisation of the rigid inverted flag by considering the pivot point to occur at the Cartesian position, $x_0$, along the plate; $x_0 = -L$ is the leading edge of the plate, with $x_0 = 0$ being its trailing edge, as per figure 1. The resulting hydrodynamic torque per unit width, $\boldsymbol {M}_{hydro}$, experienced by the plate about its pivot point, $x_0$, is given by thin airfoil theory in the inviscid limit (Tchieu & Leonard Reference Tchieu and Leonard2011)

(4.1)\begin{equation} \boldsymbol{M}_{hydro} ={-} \tfrac{1}{2} \rho U^2 L^2 C_M \hat{\boldsymbol{z}}, \end{equation}

where $\hat {\boldsymbol {z}}$ is the Cartesian basis vector in the $z$-direction (see figure 1), and the dimensionless torque coefficient is

(4.2)\begin{equation} C_M = 2 {\rm \pi}\theta \left(\frac{x_0}{L}+\frac{3}{4}\right). \end{equation}

Adding this torque per unit plate width to that produced by the torsional spring, $\boldsymbol {M}_{spring} = k_\theta \hat {\boldsymbol {z}}$, at the pivot point, gives the critical dimensionless flow speed for a divergence (zero stiffness) instability

(4.3)\begin{equation} {\kappa}_{{critical}} = \frac{\rho U^2 L^2}{k_\theta} = \frac{1}{\rm \pi}\left(\frac{x_0}{L}+\frac{3}{4}\right)^{{-}1}, \end{equation}

establishing that a divergence instability occurs only if the pivot position satisfies

(4.4)\begin{equation} x_0 >{-}\frac{3 L}{4}, \end{equation}

i.e. the pivot position must be downstream from the quarter-chord position. This shows that a rigid conventional flag (with a pivot position at its leading edge, $x_0 = -L$) cannot exhibit a divergence instability; it also cannot undergo coupled-mode flutter (at least due to coupling between structural modes) due to its single structural degree of freedom.

For the rigid inverted flag considered here, i.e. $x_0 = 0$, (4.3) gives a divergence instability at

(4.5)\begin{equation} {\kappa}_{{critical}} = \frac{4}{3{\rm \pi}} \approx 0.424. \end{equation}

The precise boundary between states I and II in the simulations is difficult to discern, but certainly occurs for $\kappa > {\kappa }_{{critical}} \approx 0.424$. This critical (inviscid) value is indicated in figure 3 as a dashed vertical line. Finite Reynolds number is known to lower the lift experienced by a thin airfoil (Gurugubelli & Jaiman Reference Gurugubelli and Jaiman2015), which is consistent with this enhanced viscous value for $\kappa _{critical}$ in simulations. This observation suggests that the rigid inverted flag undergoes a divergence instability as the dimensionless flow speed, $\kappa$, is increased, following which (deflected) state II emerges and is observed for all angles, $\theta \lesssim 5.7^\circ$. This behaviour is similar to that observed for the flexible inverted flag, where a small-amplitude deflected equilibrium is observed for low flow speeds (Goza et al. Reference Goza, Colonius and Sader2018).

4.1.3. Small-amplitude simple flapping: state III

As the flow speed, $\kappa$, is increased further, state II gives way to unsteady motion of the rigid inverted flag. Figure 5 shows such a ‘state III’ for a slightly larger value of $ {\kappa } = 0.713$ with respect to the case shown for state II ($ {\kappa } = 0.675$). Very small-amplitude periodic oscillations are observed to occur in state III about the mean deflected position; the flag does not cross its zero deflection equilibrium position, $\theta = 0$. For the flexible inverted flag, Goza et al. (Reference Goza, Colonius and Sader2018) showed that this state arises from a bifurcation of the small-amplitude deformed equilibrium.

Figure 5 shows that the wake of state III is still dominated by long shear layers, but these layers start to break up downstream. This state bears a striking resemblance to small-amplitude stall flutter (Razak, Andrianne & Dimitriadis Reference Razak, Andrianne and Dimitriadis2011). Physically, such an instability occurs via the following mechanism:

1. (i) Initially, the deflected flag generates enough lift to deflect it beyond the angle at which the flow separates from its leading edge and upper surface.

2. (ii) This flow separation causes a loss of lift, which leads to the flag angle decreasing.

3. (iii) The decreased angle allows the flow to reattach to the flag surface.

4. (iv) The flag returns to its original position and the process repeats.

Because the flag angle is always small, the separated flow does not roll up into a distinct vortex but merely deforms the shear layers in the wake.

While the explanation of the observed behaviour in terms of stall flutter is not definitive, it appears that small-amplitude stall flutter could also describe the observed small-amplitude periodic oscillation in the flexible inverted flag (Goza et al. Reference Goza, Colonius and Sader2018), discussed above. Note the similarity in the Strouhal numbers exhibited by the rigid and flexible inverted flags in this regime: for a light flexible inverted flag, Goza et al. (Reference Goza, Colonius and Sader2018) record a value of $St \equiv fL/U \approx 0.1$, whereas the example shown in figure 5 for a light rigid inverted flag with $\mu = 4.54$ and $\kappa = 0.713$ gives $St \approx 0.09$.

4.1.4. Small-amplitude complex flapping: state IIIA

The presence of stall flutter is further supported by the following observation, where the dimensionless flow speed is slightly increased from $ {\kappa } = 0.713$ (for state III) to $ {\kappa } = 0.733$. This also leads to small-amplitude flapping but now with a complex time series at significantly lower fundamental frequency, and around the zero deflected equilibrium, $\theta = 0$; see figure 5. The wake is again formed by two shear layers which are deformed/modulated by the flag motion. Strikingly, the frequency spectrum shows a primary oscillation frequency approximately one third that of state III. The complex and distorted time series of $\theta$, relative to state III (at $ {\kappa } = 0.713$), indicates the origin of this behaviour:

1. (i) Initially, there is one cycle of the stall flutter process, that increases the deflection angle then decreases it following separation. However, the maximum deflection angle exceeds that of state III which leads to a very slight overshoot past $\theta = 0$.

2. (ii) The flag then re-approaches $\theta = 0$, which is unstable once the flow reattaches, because $\kappa$ exceeds that for divergence instability.

3. (iii) The stall flutter cycle repeats, but now on the other side of $\theta = 0$.

The first and third steps in this process are identical to that for state III, while the time series in figure 5 shows that the second step takes similar time. This explains the origin of the threefold decrease in the observed oscillation frequency relative to state III.

Other examples of this small-amplitude complex flapping state are observed for different mass ratios, $\mu$ (data not shown). They contain slightly different frequency profiles and some exhibit oscillations about $\theta \neq 0$. However, all of this dynamics can be described by the above-mentioned stall flutter process. State IIIA, with its complex time series, has not been reported in previous studies of flexible inverted flags; it occurs over a very small range of $\kappa$.

The stability analysis of Goza et al. (Reference Goza, Colonius and Sader2018), for the flexible inverted flag, indicates that the small-deflection flapping state is a limit cycle oscillation that encircles the unstable equilibrium position. The numerical results here, for the rigid inverted flag, show that the oscillation amplitudes of states III and IIIA are similar. Moreover, states III and IIIA do not significantly alter the vorticity and wake structure near the body, with no coherent vortex shedding; see figure 5. Taken together, these facts suggest that state IIIA results from a heteroclinic connection between the two (now unstable) limit cycles on each side of $\theta = 0$ for state III.

This conclusion is further supported by the results in figure 6, which gives the trajectories in $(\theta, \dot {\theta })$ phase space, for $\kappa = 0.713$ (state III) and $\kappa = 0.733$ (state IIIA). Note that the deflection angle never crosses $\theta = 0$ for state III, while it overshoots $\theta = 0$ for state IIIA. Once the trajectory crosses $\theta = 0$, it is initially attracted into the limit cycle traced by state III (at very similar $\kappa$). The trajectory then spirals out, gradually diverging from the state III limit cycle and causing the angle to again cross $\theta = 0$ in the opposite direction. The same process is repeated on the other side, completing the heteroclinic cycle.

Figure 6. Phase portraits in the $(\theta, \dot {\theta })$ space of the small-deflection flapping state III and the small-amplitude flapping state IIIA. Blue solid/dashed lines show limit cycles for the small-deflection flapping state deflected to the left/right, for $\mu = 4.54$ and $\kappa = 0.713$. The fine orange line is the cycle for the small-amplitude flapping state IIIA with a small increase in $ {\kappa }$, to $ {\kappa } = 0.733$. For the state IIIA case, when the angle crosses $\theta = 0$ (i.e. the flag overshoots the neutral position) a path is traced that spirals away from the limit cycle for the state III case, which causes the flag to overshoot in the other direction; the process repeats on the other side of the neutral position.

4.2.1. Interaction between vortex shedding and flag natural frequencies

Knisely (Reference Knisely1990) studied vortex shedding by rigid flat plates and showed that the Strouhal number, $St \equiv f_{vs}H/U \approx 0.17$, for all appreciable angles of attack $\phi$, where $f_{vs}$ is the vortex shedding frequency and $H$ is the projected frontal length. For the rigid inverted flag, this frontal length, $H$, is directly related to the plate length, $L$, and the amplitude of oscillation, $\theta$, via $H = 2L\sin \theta$. Therefore, the vortex shedding frequency normalised by the plate length can be estimated in the quasi-static limit, as $f_{vs}L/U \approx 0.085/\sin \theta$. Figure 3 shows that at the onset of large-amplitude flapping, the amplitude is $\theta \approx 45 {^\circ }$, which gives a normalised vortex shedding frequency $f_{vs}L/U \approx 0.12$. This frequency and its subharmonic are marked on the spectrograms of figure 3.

We now calculate an estimate for the natural flapping frequency for the rigid inverted flag. Its natural frequency in vacuo, $f_N = 1/(2{\rm \pi} ) \sqrt {k_\theta /I}$, where $I$ is the moment of inertia per unit width of the flag, is not appropriate here due to the significant added mass (inertia) of the fluid. This added inertia, characterised by its moment per unit width, $I_a$, is proportional to that of a cylinder of fluid of radius, $L$, i.e. $I_{{cyl}}$, that is the potential volume swept by the flag, i.e.

(4.6)\begin{equation} I_a = c_I I_{{cyl}}, \end{equation}

where $c_I$ is a dimensionless order-one coefficient. We can non-dimensionalise $I_{{cyl}}$ by the flag's moment of inertia, $I$, to obtain

(4.7)\begin{equation} \frac{I_{{cyl}}}{I} = \frac{\dfrac{1}{2}{\rm \pi}\rho L^4}{\dfrac{1}{3}\rho_s h L^3} = \frac{3}{2}{\rm \pi}\mu. \end{equation}

Combining (4.6) and (4.7) gives

(4.8)\begin{equation} I_a = \tfrac{3}{2}{\rm \pi} c_I \mu I, \end{equation}

which then leads to the corrected natural frequency of the rigid inverted flag

(4.9)\begin{equation} f_{N_{f}} = \frac{1}{2{\rm \pi}}\sqrt{\frac{k_\theta}{I+I_a}} = f_N\sqrt{\frac{1}{1+ \dfrac{3}{2} {\rm \pi}c_I \mu}}, \end{equation}

where $c_I$ remains to be evaluated.

Note that this is essentially the same idea as that traditionally employed when calculating an added mass to analyse the VIV of circular cylinders (Gopalkrishnan Reference Gopalkrishnan1993; Khalak & Williamson Reference Khalak and Williamson1996; Govardhan & Williamson Reference Govardhan and Williamson2002). The ‘impulsive added mass’ – derived from the hydrodynamic force arising from impulsive motion of the body – can be calculated (Magnaudet Reference Magnaudet2011), and this has been done with some success for oscillating cylinders that are both externally controlled and undergoing VIV (Leonard & Roshko Reference Leonard and Roshko2001; Carberry, Sheridan & Rockwell Reference Carberry, Sheridan and Rockwell2005). An ‘apparent added mass’ could also be derived from all of the fluid force that is in phase with the acceleration, which will include the impulsive added mass plus a component due to the presence of vorticity. In the remainder of this section and Appendix B, we determine $c_I$ both theoretically in the inviscid limit, and empirically by fitting to the simulation data thereby including the impact of viscosity and vorticity.

No impinging flow: we first consider the special case of no impinging flow and small-amplitude oscillation, which is amenable to both analytical and numerical evaluation. This provides a baseline for the effect of non-zero impinging flow. The evaluation is characterised in terms of the oscillatory Reynolds number

(4.10)\begin{equation} Re_{\omega} = \frac{\omega_{N_{f}}L^2}{\nu}, \end{equation}

where the angular frequency, $\omega _{N_{f}}= 2{\rm \pi} f_{N_{f}}$. The case of inviscid flow, $Re_{\omega } \rightarrow \infty$, is calculated analytically in Appendix B and gives

(4.11)\begin{equation} c_I = \tfrac{9}{64} \approx 0.141. \end{equation}

To evaluate $c_I$ for finite $Re_{\omega }$, numerical simulations are performed using the immersed-boundary code for the flag with a very small initial displacement. The resulting damped oscillations or ‘ring down’ then provides access to the linear resonant frequency and hence $c_I$ (Chakraborty et al. Reference Chakraborty, Van Leeuwen, Pelton and Sader2013). These numerical results are reported in figure 22, showing that the effect of viscosity is to enhance the added mass (inertia) in the fluid, as expected.

Large-amplitude flapping: next, we consider the large-amplitude flapping dynamics of a rigid inverted flag, for which we determine $c_I$ using (4.9) (Chakraborty et al. Reference Chakraborty, Van Leeuwen, Pelton and Sader2013); $f_{N_{f}}$ is taken to be the flapping frequency. Figure 3 shows that for the lighter flags ($\mu = 2.27$ and $4.54$), large-amplitude flapping first occurs via a periodic state $ {P_{1}}$ that is synchronised with the vortex shedding frequency. Figure 7 gives $c_I$ for these cases as a function of the dimensionless flow speed, $\kappa$, and the oscillatory Reynolds number $Re_\omega$. The reported values of $c_I$ are an order-of-magnitude higher than for the case of no impinging flow (above), commensurate with substantial vortex shedding in the $ {P_{1}}$ state. Interestingly, $c_I$ is found to be a weak function of $\kappa$ which ranges from $c_I \approx 1.4$ at the onset of flapping down (lower $\kappa$) to $c_I \approx 1.2$ at the loss of stability of the periodic state (higher $\kappa$).

Figure 7. Added inertia coefficient, $c_I$, defined in (4.9) as a function of (a) the dimensionless flow speed, $\kappa$, and (b) an oscillatory Reynolds number, $Re_{\omega }$, for the first periodic flapping state $ {P_{1}}$ identified for the two mass ratios, $\mu = 4.54$ ($\bullet$, blue) and $\mu = 2.27$ ($\bullet$, orange).

As a heuristic, we apply the value $c_I \approx 1.4$ obtained near the onset of $ {P_{1}}$ flapping, to all flapping states to calculate a natural frequency $f_{N_f}$ accounting for the apparent added mass, shown as the solid curve in the spectrograms of figure 3. This provides reasonable agreement with the dominant frequencies in the numerically evaluated spectrograms, suggesting that inertia induced by vortex shedding is the primary mechanism controlling the flapping frequency, at the onset of flapping.

Finally, these estimates of vortex shedding frequency, $f_{vs}$, and the natural frequency in fluid, $f_{N_{f}}$, provide a framework to understand the appearance of stable periodic states as a synchronisation between these two frequencies. The first periodic state may be expected when $f_{vs}$ and $f_{N_{f}}$ coincide. Figure 3 shows that the value of $ {\kappa }$ at which this coincidence occurs is quite close to the value at which the periodic state $ {P_{1}}$ occurs with the onset of large-amplitude flapping. However, for the heavier flag ($\mu = 0.39$), $f_{vs} = f_{N_{f}}$ at a value of $ {\kappa }$ that is much lower, or equivalently, where the flag natural frequency is lower. In this case, the vortex shedding is too fast to match the preferred frequency of the flag, and the synchronisation to the periodic $ {P_{1}}$ does not occur. However, for this same heavier flag the value of $ {\kappa }$ at which the first subharmonic of the vortex shedding frequency coincides with the natural frequency ($f_{vs}/2 = f_{N_{f}}$) is also lower, and a large range of $ {\kappa }$ is observed where the flow synchronises to a subharmonic state $ {P_{2}}$.

The various organising states can therefore be thought of as a synchronisation between the vortex shedding at a preferred frequency, and the oscillation of the plate at a corrected natural frequency. This natural frequency is that which incorporates the impact of the vortices on the apparent added mass. There is a type of ‘basin of attraction’ for this to occur, and the two frequencies need to be reasonably close for nonlinear effects to synchronise them. This suggests that even in situations where these synchronised states may not be stable, they can still play a role in the dynamics when the flow is chaotic. We show in § 4.3 that the chaotic oscillations are not the product of a single mechanism. There are multiple routes to chaos depending on the value of $ {\kappa }$ and $\mu$, which are governed by the periodic states (albeit unstable) outlined above.

4.2.3. Synchronised state: $ {P_{1}}$

Figure 8 presents data for the first organising periodic (synchronised) state, $ {P_{1}}$, with $ {\kappa } = 0.835$ and $\mu = 4.54$. The time history and frequency spectrum both show clear periodicity, with an amplitude of $ {\theta _{{max}}} \approx 45 {^\circ }$. Vortex shedding is also periodic with the same frequency as the flag oscillation. The wake structure is reasonably complex with the interaction of vortices generated from the leading and trailing edge of the flag. Goza et al. (Reference Goza, Colonius and Sader2018) identified a synchronised state as the first sub-regime of their mode IV for the flexible inverted flag. State $ {P_{1}}$ matches the ‘VIV mode’ identified by Goza et al. (Reference Goza, Colonius and Sader2018); we have renamed it to fit an overall classification of a more extensive collection of states.

Figure 8. Synchronised state $ {P_{1}}$: (a) time history of the angular displacement; (b) frequency spectrum of the angular displacement; and (c) an image of the vorticity field at $\tau = 350$, for $ {\kappa } = 0.835$ and $\mu = 4.54$. Black dashed line marks the nominal Strouhal frequency $f = 0.12$. Orange dashed line marks the first subharmonic, $f = 0.06$.

Vortex formation process: the far wake consists of four distinct vortices per cycle: a pair of oppositely signed strong vortices in the middle of the wake and a weaker vortex of each sign off to each side. The strong vortices in the middle are organised in an inverted Kármán configuration, i.e. their sense of rotation induces a jet, not a wake. However, the structure nearer the body shows that six unique vortices are shed per oscillation cycle, or three per half-cycle. Two of these vortices are formed as a pair due to motion of the leading edge of the flag, as it accelerates from close to its maximum displacement back towards its neutral position; they are shed close to the point where the flag is pointing straight into the flow, both ejected from the trailing edge. The pair of vortices are annotated as $L_{1}^-$ and $L_{1}^+$ in figure 8: $L_{1}^-$ is the leading-edge vortex formed by the separated flow at the leading edge as it accelerated down; $L_{1}^+$ is a vortex of opposite sign formed from the vorticity on the top of the flag that is induced by the formation of the separated leading-edge vortex $L_{1}^-$. The third vortex, $T_{1}^-$, is shed from the trailing edge when the flag is close to its maximum anti-clockwise displacement presenting an upwards ramp to the flow. This process occurs in reverse as the flag accelerates up; the three vortices, $L_{2}^+$, $L_{2}^-$ and $T_{2}^+$, shed during the previous upward-accelerating half-cycle are also annotated on figure 8. The stronger vortex of the leading edge pair eventually wraps the weaker vortex and annihilates it, forming a single large central wake vortex, whereas the trailing-edge vortex becomes the weaker vortex off to the side of the wake.

4.2.4. Asymmetric subharmonic state: $ {P_{2_{A}}}$

The second of the organising states is a large-amplitude state which is asymmetric in terms of the lift and mean displacement of the flag. An example of this subharmonic state is shown in figure 9, for $ {\kappa } = 1.013$ and $\mu = 4.54$. Its periodicity is clear from the displacement time history, and its subharmonic nature is evident by the significant frequency component at half of the primary frequency.

Figure 9. Asymmetric state $ {P_{2_{A}}}$: (a) time history of the angular displacement; (b) frequency spectrum of the angular displacement; and (c) an image of the vorticity field at $\tau = 300$, for $ {\kappa } = 1.013$ and $\mu = 4.54$. Black dashed line marks the nominal Strouhal frequency, $f = 0.12$. Orange dashed line marks the first subharmonic, $f = 0.06$.

Vortex formation process: figure 10 shows a sequence of images over one cycle of oscillation, along with the time history of the flag displacement over two cycles of oscillation. The first two images show that as the flag accelerates from its maximum clockwise displacement to its neutral position, a large negative leading-edge vortex is formed. This pairs with the vortex formed by the positive vorticity induced on the top surface of the flag, forming the first leading-edge pair ($L_-^1, L_+^1$). The third and fourth images of figure 10 show that as the flag accelerates from its maximum anti-clockwise displacement towards its neutral position, a similar process occurs but with reversed signs on the vortices, forming the second leading-edge pair ($L_+^2, L_-^2$). However, as the flag approaches its neutral position it does not cross the centreline, and it instead stagnates at a small negative angle. During a period of stagnation (see fifth image), a small pair of vortices are shed into the wake ($W_+, W_-$) – essentially from the trailing edge of the flag – and the flag deflects in the anti-clockwise direction again. After this anti-clockwise deflection, the flag then again accelerates towards its neutral position as shown in the sixth image, and during this acceleration a final pair of leading-edge vortices are formed and shed ($L_+^3, L_-^3$). The flag motion continues past the neutral position to its maximum clockwise deflection, and the cycle repeats.

Figure 10. Asymmetric state $ {P_{2_{A}}}$: snapshots of vorticity in the near wake, with the time history of angular displacement of the flag. Circles on the time history coincide with the instants shown in the images. One leading-edge pair of vortices, ($L_-^1, L_+^1$), is shed from the top side of the flag, while two leading-edge pairs, ($L_-^2, L_+^2$), ($L_-^3, L_+^3$), are shed from the bottom side, with a wake pair, ($W_+, W_-$), shed in between these two leading-edge pairs. The example shown is the same case as figure 9, i.e. $ {\kappa } = 1.013$ and $\mu = 4.54$.

4.2.5. Symmetric subharmonic state: $ {P_{2}}$

The third organising state is a subharmonic state, with two cycles of vortex shedding occurring over a single cycle of the flag motion. Figure 11 shows an example case of this state for $ {\kappa } = 1.572$ and $\mu = 0.39$, which is only stable at this lower $\mu$. The subharmonic nature of the state is clear from the spectrum in figure 11: the primary peak is aligned with $f_{vs}/2$. Its amplitude is very large, with the angle of displacement exceeding $90 {^\circ }$, so that the flag goes past a vertical orientation; see flow visualisation in figure 12. The flag spends a large portion of the cycle approximately normal to the flow, which dictates the vortex shedding and formation.

Figure 11. Symmetric subharmonic state $ {P_{2}}$: (a) time history of the angular displacement; (b) frequency spectrum of the angular displacement; and (c) an image of the vorticity field at $\tau = 300$, for the case $ {\kappa } = 1.572$ and $\mu = 0.39$. Black dashed line marks the nominal Strouhal frequency, $f = 0.12$. Orange dashed line marks the first subharmonic, $f = 0.06$.

Figure 12. Close-up images of the symmetric subharmonic state $ {P_{2}}$ over a half-cycle of oscillation. (a) At maximum displacement, $\theta = 98 {^\circ }$; (b) during the acceleration phase, with the flag at $\theta = 61 {^\circ }$; (c) at the neutral position, pointing almost straight into the flow $\theta = 5 {^\circ }$; (d) during the deceleration phase, $\theta = -52 {^\circ }$; and (e) at maximum displacement pointing downward, $\theta = -98 {^\circ }$. Three main vortices – a Kármán wake pair, ($k_-$ and $k_+$), and a leading-edge vortex, ($L_-$) – dominate the dynamics, with a smaller vortex induced by the leading-edge vortex of opposite sign, ($L_+$), and an elongated trailing-edge vortex, $T_-$, playing a secondary role.

Vortex formation process: vortex formation and shedding is dominated by two processes: (i) Kármán-like vortex shedding when the flag is approximately normal to the flow, resulting in a vortex being shed from the leading edge, then one from the trailing edge; and (ii) formation of a leading-edge vortex during the accelerating phase of the motion. Figure 12 shows close-up images of the flow over a half-cycle of oscillation for the same case as figure 11.

The first image is at a phase of the flag motion that is close to its maximum clockwise displacement, i.e. $\theta \approx 98 {^\circ }$, and therefore is almost normal to the flow. At this phase, vortices roll up from each side of the normal plate, reminiscent of Kármán vortex shedding. The first image shows the formation of the first of the Kármán vortices of a pair from the leading edge of the plate, labelled $k_-$.

The second image shows the flag at $\theta \approx 61 {^\circ }$, during its anti-clockwise motion towards its neutral position. The second Kármán vortex forming from the trailing edge is clear (labelled $k_+$). Both the Kármán wake vortices $k_-$ and $k_+$ have grown, and there is also a large amount of negative vorticity generated from the accelerating leading edge. This is still feeding the negative Kármán vortex, but is also starting to roll up into a defined leading-edge vortex as described in the third image.

The third image shows the flag at close to its neutral position ($\theta \approx 5 {^\circ }$), pointing almost straight into the oncoming flow. The negative Kármán vortex, $k_-$, has been shed and the positive Kármán vortex, $k_+$, is about to detach with only a thin layer of positive vorticity from the lower side of the flag still feeding it. The negative vorticity previously feeding $k_-$ has rolled up into a leading edge vortex, $L_-$. This vortex, $L_-$, has progressed to the trailing edge and its presence above the flag has induced positive vorticity on the upper side of the flag.

The fourth image shows the flag at $\theta \approx -52 {^\circ }$. The negative Kármán vortex, $k_-$, has been shed while the positive Kármán vortex, $k_+$, and the negative leading-edge vortex, $L_-$, have been ejected from the trailing edge as a dipole pair. The detachment and shedding of the negative leading-edge vortex, $L_-$, and the fact that the flag top surface when angled down generates a favourable pressure gradient, induces the flow to reattach. Consequently, the positive vorticity that had been formed on the top surface rolls up into a small positive vortex, $L_+$, trailing the larger dipole pair. The negative vorticity formed in the boundary layer on the top surface of the flag also starts to roll up into a vortex labelled, $T_-$, as it separates from the trailing edge. These small positive and negative vortices are secondary to the main Kármán vortices, $k_-$ and $k_+$, and the leading-edge vortex, $L_-$.

The fifth and final image shows the flag at close to its maximum anti-clockwise displacement. The vortex structure is a mirror image of the structure at maximum clockwise displacement.

4.2.6. Deflected state: $ {D}$

The final state is the deflected state, $ {D}$, shown in figure 13 for $ {\kappa } = 2.913$ and $\mu = 4.54$. This is almost stationary in a laid-over configuration, where the hydrodynamic force experienced by the flag is balanced by the torsional spring. The wake structure and vortex shedding frequency is similar to that of a stationary flat plate. This state coincides with mode VI for the flexible inverted flag identified by Goza et al. (Reference Goza, Colonius and Sader2018). While vortex shedding from state $ {D}$ is periodic in the near wake, there is a secondary instability that leads to the two rows of Kármán vortices breaking down into a lower frequency, longer wavelength configuration. In figure 13, this occurs around $10L$ downstream. This phenomenon has been documented in a number of bluff-body wakes that form this two-row vortex structure, e.g. see Hosseini, Griffith & Leontini (Reference Hosseini, Griffith and Leontini2021), and is associated with an absolute instability of the mean wake profile.

Figure 13. Deflected state $ {D}$: (a) time history of the angular displacement; (b) frequency spectrum of the angular displacement; and (c) an image of the vorticity field at $\tau = 342$, for the case $ {\kappa } = 2.815$ and $\mu = 4.54$. Black dashed line marks the nominal Strouhal frequency $f = 0.12$. Orange dashed line marks the first subharmonic $f = 0.06$.

4.3.1. Instability of the synchronised state $ {P_{1}}$: type I intermittency

The first route to chaos that we examine is the loss of stability of state $ {P_{1}}$. Figure 14 shows time history of the angular displacement for (i) a stable $ {P_{1}}$ case, and (ii) an unstable case with a slightly higher value of $ {\kappa }$. The latter case shows almost periodic oscillations, interspersed with high-amplitude bursts. Both cases are also plotted in the $(\theta, \dot {\theta })$ phase space which shows a limit cycle for the stable case that appears to act as an attractor; orbits of the unstable case are close to this attractor for a number of cycles (with almost periodic oscillations), but eventually spiral away (high-amplitude bursts), before returning back towards the attractor. This ‘type I intermittency’ (Pomeau & Manneville Reference Pomeau and Manneville1980) has been demonstrated in a number of dynamical systems and discrete maps.

Figure 14. Instability of $ {P_{1}}$ leading to type I intermittency. (a,b) Show time histories of flag angular displacement for mass ratio, $\mu = 2.27$, with $\kappa = 0.810$ ((a), blue) and $\kappa = 0.885$ ((b), orange). These represent a stable $ {P_{1}}$ case, and an unstable case. The same cases are plotted in the $(\theta, \dot {\theta })$ plane on the right; the stable case with a bold blue line and the unstable case with the finer orange line. This plot shows the limit cycle for the stable case, with the unstable case orbiting close to this limit cycle for many periods before spiralling away, then collapsing back to the unstable limit cycle.

We note here that this route to chaos only relies on a periodic solution existing at nearby parameter values. The periodic solution itself may never be observed. We highlight this point as this type I intermittency occurs for both higher and low values of $\mu$, while a stable periodic solution (state $ {P_{1}}$) is only observed for the higher-$\mu$ cases. The fact that $ {P_{1}}$ is not observed for the lower-$\mu$ case, $\mu = 0.39$, does not preclude this route to chaos.

4.3.2. Instability of state $ {P_{2_{A}}}$ generates heteroclinic-like orbits

The second route to chaos is organised around state $ {P_{2_{A}}}$. Because this state is asymmetric, it can be biased to either positive or negative angular displacements. This presents two attracting states, either of which can be reached depending on the initial conditions if the state is stable. If this $ {P_{2_{A}}}$ state is unstable, it presents two unstable limit cycles in the $(\theta, \dot {\theta })$ phase space. The flag dynamics may visit each of these cycles for some time, before diverging and potentially settling towards the other cycle. After some time the dynamics will diverge from this cycle, and settle again towards the original cycle, where the process can repeat. This is similar to a heteroclinic cycle – the cyclic nature of the dynamics is driven by attraction to one state, but then repulsion in at least one direction in phase space. This direction of repulsion from one state drives the dynamics towards the second state, which itself is unstable – which drives the dynamics back towards the original state.

Figure 15 shows an example of this process. In panel (a), the two possible limit cycles in the $(\theta,\dot {\theta })$ phase space are shown for a stable $ {P_{2_{A}}}$ . In panel (b), the trajectory traced by a case at a slightly higher value of $ {\kappa }$, which is unstable, is given. This clearly shows that the unstable orbit spends time cycling near an orbit of the $ {P_{2_{A}}}$ state, before spiralling away and being attracted toward the orbit of the $ {P_{2_{A}}}$ state biased to the other side. The cycle is not perfect, leading to a chaotic trajectory organised around the two manifestations of the $ {P_{2_{A}}}$ state.

Figure 15. Phase portraits in the $(\theta, \dot {\theta })$ plane, demonstrating heteroclinic-like mode competition between the two manifestations of the asymmetric state $ {P_{2_{A}}}$. (a) Shows the trajectory for state $ {P_{2_{A}}}$, at $\mu = 2.27$ and $\kappa = 0.996$. Solid/dashed line represents the state biased to the left/right. (b) Shows the trajectory for a case at a very slightly higher, $\kappa = 1.013$. The image shows that the dynamics spends time near one orbit before losing stability and spiralling to the orbit biased to the other side.

4.3.3. Mode competition between states $ {P_{2_{A}}}$ and $ {P_{2}}$

The third route to chaos is apparent mode competition between states $ {P_{2_{A}}}$ and $ {P_{2}}$. Similar to the other states, this is demonstrated by trajectories in the $(\theta,\dot {\theta })$ phase space. Figure 16 shows (left to right) (i) the trajectories of the two possible, stable $ {P_{2_{A}}}$ states at $\mu = 2.27$ and $ {\kappa } = 0.996$, then (ii) the cycle for the stable $ {P_{2}}$ state (albeit at a lower value of $\mu = 0.39$), followed by (iii) the chaotic trajectory at $\mu = 2.27$ and $ {\kappa } = 1.822$.

Figure 16. Phase portraits in the $(\theta, \dot {\theta })$ plane illustrating mode competition between organising states $ {P_{2_{A}}}$ and $ {P_{2}}$. (a) The trajectories for the stable $ {P_{2_{A}}}$ state, at $\mu = 2.27$ and $\kappa = 0.996$. Solid/dashed line represents the state biased to the left/right. (b) The trajectory for the stable $ {P_{2}}$ state, at $\mu = 0.39$ and $\kappa = 1.572$. (c) The trajectory for the chaotic oscillation, at $\mu = 2.27$ and $\kappa = 1.822$; the chaotic trajectory spends periods orbiting near each of the organising states.

The $ {P_{2}}$ state is not stable for higher-$\mu$ flags, and this is why the example cycle shown in figure 16 is from the lower-$\mu$ flag. Note, however, that the chaotic trajectory does seem to orbit near this example cycle, providing evidence that the $ {P_{2}}$ state is still playing a role in organising the dynamics for the higher-$\mu$ flags.

4.3.4. Mode competition between states $ {P_{2}}$ and $ {D}$

The fourth route to chaos is apparent mode competition between the unstable $ {P_{2}}$ state and the two manifestations of the deflected state $ {D}$ (deflected either to positive or negative angular displacement). Figure 17 gives the trajectory of an example case compared with stable examples of the two organising states involved. The range of $\dot {\theta }$ covered by the chaotic trajectory is smaller than that for the example where $ {P_{2_{A}}}$ plays a role (figure 16). Moreover, a number of small loops appear that gradually spiral out near the extremities of the range of $\theta$ covered. This shows that the flag spends long portions of time near the deflected $ {D}$ state, which eventually loses stability, causing the flag to execute oscillations similar to the subharmonic $ {P_{2}}$ state, before settling towards the $ {D}$ state again, possibly biased to the other side.

Figure 17. Phase portraits in the $(\theta, \dot {\theta })$ plane illustrating mode competition between organising states $ {P_{2}}$ and $ {D}$. (a) The trajectory for the stable $ {P_{2}}$ state, at $\mu = 0.39$ and $\kappa = 1.572$. (b) The trajectory for the stable $ {D}$ state, at $\mu = 2.27$ and $\kappa = 2.545$. Solid/dashed line represents the state biased to the left/right. (c) The trajectory for the chaotic oscillation, at $\mu = 2.27$ and $\kappa = 2.169$; the chaotic trajectory spends periods orbiting near each of the organising states.

4.3.5. Instability of the deflected state $ {D}$

The fifth and last route to chaos found for all flags is mode competition between the regions of the deflected state $ {D}$, i.e. positive and negative angular displacement. Figure 18 gives the trajectory of an example case, along with the two stable limit cycles representing $ {D}$. This shows that the trajectory is dominated by loops that orbit near one of the $ {D}$ cycles, that slowly spiral out. Physically, this represents the flag being deflected almost normal to the flow and exhibiting small, but growing oscillations. Eventually these oscillations become so large that they cause the flag to overshoot the centreline of the flow, causing it to be deflected to the other side, where this cycle again repeats.

Figure 18. Phase portraits in the $(\theta, \dot {\theta })$ plane illustrating mode competition between the two possible manifestations of organising state $ {D}$. (a) Shows the trajectory for the stable $ {D}$ state, at $\mu = 2.27$ and $\kappa = 2.545$. Solid/dashed line represents the state biased to the left/right. (b) Shows the trajectory for the chaotic oscillation, at $\mu = 2.27$ and $\kappa = 2.291$; the chaotic trajectory spends periods orbiting near each of the examples of state $ {D}$, before flipping to the other.

This final route to chaos appears to be the state hypothesised by Goza et al. (Reference Goza, Colonius and Sader2018) to drive the chaotic oscillations in the flexible inverted flag. Whether the other routes to chaos identified above are present for the flexible inverted flag remains an open question.

4.3.6. Development of chaos across the flapping regime for higher $\mu$

The changing nature of the chaotic dynamics is further investigated in figure 19, which shows the time histories of the angular displacement for increasing $ {\kappa }$ with $\mu = 2.27$.

Figure 19. Development of the time history of oscillation with increasing $ {\kappa }$, for $\mu = 2.27$ in the large-amplitude flapping range. For low $ {\kappa }$ (e.g. $ {\kappa } = 1.131$), the flag oscillates for long epochs in a state similar to $ {P_{1}}$, interspersed with intermittent losses of stability. For slightly higher $ {\kappa }$ (e.g. $ {\kappa } = 1.312$), there are epochs of oscillation similar to $ {P_{2_{A}}}$, first biased to one side then the other. Higher $ {\kappa }$ again (e.g. $ {\kappa } = 1.506$) shows intermittent switching between epochs similar to $ {P_{2_{A}}}$ and $ {P_{2}}$. Higher $ {\kappa }$ again (e.g. $ {\kappa } = 1.822$) shows intermittent switching between epochs similar to $ {P_{2}}$ and the deflected state D. Finally for the highest $ {\kappa }$ shown, the dynamics settles to a stable deflected state D periodic oscillation.

For the smallest flow speeds, i.e. $ {\kappa } < 1.2$, there are reasonably long epochs of time where the flag oscillates in a state similar to the periodic state $ {P_{1}}$, interspersed amongst intermittent epochs of disordered oscillations. This is seen in figure 19 for $ {\kappa } = 1.131$ – there is a long epoch, $170 < \tau < 250$, where the flag oscillates almost periodically, but the oscillations either side of this epoch fluctuate in amplitude and frequency. This is another example of the first route to chaos shown in figure 14.

Slightly increasing $ {\kappa }$ sees the flag switch between two types of epochs: (i) the flag oscillates in a state similar to the periodic but asymmetric state $ {P_{2_{A}}}$ biased to one side, and (ii) the same state biased to the other side. See the example for $ {\kappa } = 1.312$ in figure 19, which shows that there are a number of cycles with the flag biased to negative angular displacement over $220 < \tau < 240$, and a cycle biased to positive angular displacement over $270 < \tau < 285$. This is an example of the second route to chaos illustrated in figure 15.

Increasing $ {\kappa }$ further leads to epochs similar to the asymmetric state $ {P_{2_{A}}}$, interspersed with epochs reminiscent of the subharmonic state $ {P_{2}}$. See the example for $ {\kappa } = 1.407$ in figure 19 which shows chaos via mode competition between states $ {P_{2_{A}}}$ and $ {P_{2}}$ – the third route to chaos – as outlined in figure 16.

A further increase in $ {\kappa }$ produces intermittent switching between the subharmonic state $ {P_{2}}$ and oscillations near the periodic deflected state $ {D}$. See the examples in figure 19 for $1.713 \leq {\kappa } \leq 1.934$, which also show that as $ {\kappa }$ is increased, the duration of the epochs spent in the subharmonic state $ {P_{2}}$ decreases while epochs near the deflected state $ {D}$ become more dominant. This dynamics corresponds to mode competition between states $ {P_{2}}$ and ${ {D}}$ – the fourth route to chaos – as per figure 17.

Larger values of $\kappa$ again generate long epochs where the dynamics produces oscillations near the periodic deflected state $ {D}$ biased to one side, interspersed with epochs of the same state biased to the other side. This is shown in figure 19 for cases in the range $2.050 \leq {\kappa } \leq 2.291$. The chaotic motion cycles between the two manifestations of the deflected state $ {D}$, as outlined in figure 18 – the fifth route to chaos.

Finally, for the highest values of $ {\kappa }$, the flag oscillations settle to the periodic deflected state $ {D}$, as shown for $ {\kappa } = 2.545$ in figure 19.