3.2.1 Rationale for use of Kornecki’s steady theory

The utility of using Kornecki’s steady theory, (3.2), is as follows:

1. (i) ${\it\Omega}\neq 0$ : if (3.2) predicts a bifurcation at non-zero frequency, ${\it\Omega}$ , then unsteadiness in the hydrodynamic pressure load may affect the sheet’s stability. Consequently, terms describing the unsteady hydrodynamic load, which involve the added mass parameter ${\it\mu}$ , must be included, i.e. the full unsteady theory of Kornecki et al. (Reference Kornecki, Dowell and O’Brien1976) must be used.

2. (ii) ${\it\Omega}=0$ : bifurcation at zero frequency, ${\it\Omega}=0$ , establishes that the true hydrodynamic load is steady at bifurcation. That is, the steady approximation to the hydrodynamic load in (3.2) is exact. As such, the steady and unsteady theories of Kornecki et al. (Reference Kornecki, Dowell and O’Brien1976) are identical in this case, and stability is independent of ${\it\mu}$ ; see (3.2), which then gives the exact result.

This rationale simplifies analysis by allowing the identification of a divergence instability without resort to the full unsteady theory.

3.2.2 Boundary conditions and method of solution

Equation (3.2) and its associated boundary conditions are solved using a Galerkin method that makes use of its eigenfunctions in the absence of fluid (Kornecki et al. Reference Kornecki, Dowell and O’Brien1976). Both the inverted and conventional flag problems are solved using (3.2); solution to the conventional flag is provided for reference only. The boundary conditions for the inverted flag are

where the origin of the coordinate system is at the sheet’s free end; see figure 2. The first two equations in (3.5) coincide with the usual clamp condition (at $x=1$ ), whereas the latter two give zero moment and transverse shearing force at the free end ( $x=0$ ). The boundary conditions for the conventional flag are reversed to those in (3.5) (Kornecki et al. Reference Kornecki, Dowell and O’Brien1976), with the clamp condition at $x=0$ and the free end at $x=1$ ; note that the origin for the conventional flag problem is at the sheet’s clamped end.

The deflection function, $w$ , is expressed as a linear combination of the modes, ${\it\Phi}_{n}(x)$ , for the unloaded sheet (no hydrodynamic load), i.e.

where ${\it\Phi}_{n}(x)$ satisfy the required boundary conditions (above). Substituting (3.6) into (3.2), and applying the usual orthogonality relations for ${\it\Phi}_{n}(x)$ , yields a linear eigenvalue problem for the unknown coefficients $a_{n}$ and frequency ${\it\Omega}$ ; this is solved using standard matrix techniques. Solution to this problem for a given choice of $N$ gives the required result. Convergence is sought for increasing $N$ ; $N\geqslant 3$ is found to yield results accurate to within 99.9 %. This solution method is implemented in Mathematica.

3.2.3 Results for the conventional and inverted flags

Figure 4 shows the calculated (complex) frequencies, ${\it\Omega}$ , for both the conventional and inverted flag problems, as a function of the dimensionless flow speed, ${\it\kappa}$ . Emergence of a non-zero imaginary component defines the onset of an instability (bifurcation), due to the time dependence, $\exp (-\text{i}{\it\Omega}t)$ ( $t$ is dimensionless time, i is the imaginary unit), implicit in (3.2); see Kornecki et al. (Reference Kornecki, Dowell and O’Brien1976). While results for the conventional and inverted flags exhibit some obvious similarities, the striking difference is that the conventional flag problem (figure 4 b) bifurcates at non-zero frequency (see real part, at ${\it\kappa}=15.9$ ), whereas stability of the inverted flag (figure 4 a) is lost at zero frequency when ${\it\kappa}=1.85$ (to three significant figures).

Drawing on the rationale in § 3.2.1 for use of Kornecki’s steady theory, the results in figure 4(a) prove that bifurcation, i.e. loss of stability, of the inverted flag’s zero-deflection equilibrium is a steady process (occurring at zero frequency) – it is therefore independent of the added mass parameter ${\it\mu}$ ; this is compared to measurements in § 3.4.1. This finding contrasts to the conventional flag problem, with the results of figure 4(b) establishing that unsteady fluid loading can be important at bifurcation, i.e. the full unsteady theory of Kornecki et al. (Reference Kornecki, Dowell and O’Brien1976) should be used, which depends on ${\it\mu}$ . Indeed, it is well established that bifurcation of a conventional flag is strongly dependent on ${\it\mu}$ (Kornecki et al. Reference Kornecki, Dowell and O’Brien1976; Shelley & Zhang Reference Shelley and Zhang2011).

Figure 4.

Stability of the zero-deflection equilibrium, for (a) an inverted flag and (b) a conventional flag. Dimensionless frequency, ${\it\Omega}$ (3.2) versus dimensionless flow speed, ${\it\kappa}$ are shown. A positive imaginary component indicates an unstable equilibrium.

This analysis shows that, as the flow speed increases, the inverted flag of infinite aspect ratio becomes unstable at ${\it\kappa}=1.85$ due to the steady hydrodynamic lift force balancing the elastic restoring force of the sheet – higher speeds cause it to deflect from its zero-deflection equilibrium position. The inverted flag therefore exhibits a (buckling) divergence instability (Paidoussis et al. Reference Paidoussis, Price and De Langre2010). The effect of this deflection on the sheet’s subsequent dynamics is explored in the § 4.

Next, we turn our attention to examining the effects of finite sheet aspect ratio, $H/L$ , on this instability.

3.3.1 Lifting-line theory $(H/L\gg 1)$

A wing (of finite aspect ratio) in a steady flow generates edge vortices that reduce its overall lift. Lifting-line theory accounts for this effect by replacing the wing with an infinite set of horseshoe vortices, from which the overall lift is calculated. Some important conclusions from this theory are: (i) finite aspect ratio reduces the lift slope of the wing, i.e. the rate of change in lift with respect to its angle of attack, and (ii) the angle of attack for which there is no lift is unaffected by the wing’s aspect ratio.

Lifting-line theory requires calculation of the zero-lift angle of attack, ${\it\alpha}_{0}$ . This is obtained from the deflection function of the infinite aspect ratio sheet, which is specified at the bifurcation point by (3.6), see figure 5; it is calculated from the eigenvalue problem discussed in § 3.2. The deflection function at bifurcation closely resembles that due to a line load at the sheet’s free end; this is not surprising, because the pressure load is maximum at that position on the sheet where it exhibits a square root singularity. Its zero-lift angle of attack is given by

where $A$ is the sheet’s free end displacement; the deflection function $w$ is normalised by this value, see figure 5. Increasing the sheet’s displacement, $A$ , thus linearly decreases its zero-lift angle of attack.

Figure 5.

Normalised deflection function (by its maximum value) for an inverted flag of infinite aspect ratio at its bifurcation point ${\it\kappa}=1.85$ (solid blue line), calculated using the theory of Kornecki et al. (Reference Kornecki, Dowell and O’Brien1976). Deflection function for a line load at the sheet’s free end (dashed red line), shown for comparison.

As discussed above, ${\it\alpha}_{0}$ is independent of the sheet’s aspect ratio, $H/L$ . Since the deflected sheet is at a geometric angle of attack of zero, lifting-line theory then establishes that its lift curve has (i) an identical slope to that of a flat sheet with an angle of attack ${\it\alpha}=0.578A/L$ , and (ii) a zero-lift angle of attack that is given by (3.7). The sheet has a rectangular geometry (planform), and thus dependence of its lift slope on its aspect ratio, $H/L$ , is given by the standard result from lifting-line theory (Anderson Reference Anderson1991):

where $C_{L}$ is the normalised lift coefficient, and ${\it\tau}\ll 1$ is the Glauert coefficient; ${\it\tau}$ is approximated to zero here.

Equations (3.7) and (3.8) completely specify the dependence of the hydrodynamic lift force on the sheet’s displacement, $A$ , at the bifurcation point. This assumes that the deflection function of the sheet at bifurcation is unaffected by its aspect ratio, to leading order for large aspect ratio; this is validated in the next section. Importantly, ${\it\kappa}$ is (proportional to) the ratio of the hydrodynamic lift force to the elastic restoring force of the sheet, both of which vary linearly with $A$ ; as is evident from (3.1) and (3.2). It then follows that a reduction in the lift slope by a factor of $(1+2L/H)$ (see (3.8)), increases the critical value for ${\it\kappa}$ (at which bifurcation occurs) by the same factor, i.e. a higher flow velocity is required to achieve the same lift force.

As such, the critical value for ${\it\kappa}$ at which the zero-deflection equilibrium loses its stability is

Since this result is formally valid for sheets of large aspect ratio ( $H/L$ ) only, we now examine the range of its validity by comparison to numerical results that hold for all aspect ratios – these are obtained using the vortex lattice method.

3.3.2 Vortex lattice method (arbitrary $H/L$ )

The vortex lattice method (VLM) employs a vortex sheet whose strength varies over the planform of the wing. This generalises the methodology of lifting-line theory, enabling application to wings of arbitrary aspect ratio (Anderson Reference Anderson1991). As such, VLM is used here to assess the accuracy of (3.9) which is derived for large aspect ratio.

The small-deflection theory of thin plates is used in conjunction with VLM, to numerically calculate the value of ${\it\kappa}_{\mathit{lower}}$ as a function of aspect ratio. The governing equation for the (steady) dimensionless deflection function, $w$ , is (Landau & Lifshitz Reference Landau and Lifshitz1970)

where the dimensionless Cartesian spatial coordinates $(x,z)$ are scaled by the sheet’s length, $L$ , and the dimensionless pressure, $p$ , by ${\it\rho}U^{2}A/L$ .

Importantly, the deflection function $w$ is independent of the $z$ -coordinate (see figure 2) in the limit of infinite aspect ratio (Kornecki et al. Reference Kornecki, Dowell and O’Brien1976). This also holds approximately for aspect ratios, $H/L$ , much smaller than unity, because the sheet then behaves as an Euler–Bernoulli beam (Landau & Lifshitz Reference Landau and Lifshitz1970). Since $w$ is independent of $z$ in these complementary limiting cases, we approximate the deflection function for all aspect ratios by one that is independent of $z$ ; this is expected to produce a small error in ${\it\kappa}$ , by a factor less than $1-{\it\nu}^{2}$ (Sader & White Reference Sader and White1993). Multiplying both sides of (3.10) by $w$ , integrating over the sheet’s surface and utilising its boundary conditions, then gives

where $p_{\mathit{ave}}$ is the spanwise-averaged pressure distribution, i.e. in the $z$ -direction. The average pressure, $p_{\mathit{ave}}$ , is linearly dependent on the deflection function $w$ (Anderson Reference Anderson1991), and hence ${\it\kappa}$ is independent of the displacement, $A$ , as required. Equation (3.11) is used to determine the critical value, ${\it\kappa}_{\mathit{lower}}$ , as a function of the sheet’s aspect ratio, $H/L$ .

We decouple the fluid-structure problem using a Rayleigh–Ritz approach, by expanding the deflection function:

which satisfies the clamp boundary condition for the inverted flag (at $x=1$ ). The coefficients $b_{m}$ are to be determined for a fixed number of terms, $M$ , which is increased until convergence is reached.

Use of the series expansion in (3.12) enables the hydrodynamic pressure $p$ to be calculated a priori for each individual power-law deflection function, $w_{m}(x)=(1-x)^{m}$ . This is performed numerically using VLM (Tornado 2015). Due to linearity (above), the overall average pressure distribution $p_{\mathit{ave}}$ for the complete deflection function, from these individual contributions, is then immediately determined for arbitrary $b_{m}$ . This expression for the pressure is substituted into (3.11), and the unknown coefficients, $b_{m}$ , evaluated from its stationary solution. This Rayleigh–Ritz approach ensures that the resulting value for ${\it\kappa}_{\mathit{lower}}$ is a strict upper bound (Leissa Reference Leissa1969).

Because the deflection function for infinite aspect ratio is well approximated by that of an end-loaded cantilevered sheet (see figure 5), which is a simple cubic polynomial function, the general polynomial expansion in (3.12) is expected to converge rapidly. Indeed, use of three terms only ( $M=4$ ) in (3.12) is found to yield convergence of ${\it\kappa}_{\mathit{lower}}$ to better than 99 %. The number of panels in VLM is increased systematically until a convergence of greater than 95 % is achieved for each $w_{m}(x)$ ; higher levels of convergence are limited by the significant memory requirements and available computational resources. Final values for ${\it\kappa}_{\mathit{lower}}$ are derived by extrapolating calculated results as a function of the number of VLM panels; these differ by ${<}$ 5 % from values obtained using the largest number of panels.

Results for ${\it\kappa}_{\mathit{lower}}$ , using this numerical approach, are given in table 1 as a function of aspect ratio, $H/L$ . These results establish the accuracy of (3.9), which is derived in the large aspect ratio limit. Indeed, the value of ${\it\alpha}=2.1$ for the largest aspect ratio studied, $H/L=5$ , is consistent with omission of the Glauert coefficient in (3.9), cf. (3.8) and (3.9). Glauert (Reference Glauert1926) reports a value of ${\it\tau}=0.16$ for $H/L=5$ , which decreases with a reduction in aspect ratio. While (3.9) is an asymptotic formula for large aspect ratio, the data in table 1 shows that it is also accurate for very small aspect ratio – the ${\it\alpha}$ -coefficient varies by ${\approx}$ 10 % over the range of $H/L$ studied. Use of the deflection function derived from (3.2) for infinite aspect ratio (see figure 5), rather than the Rayleigh–Ritz approach, yields results for ${\it\kappa}_{\mathit{lower}}$ that only slightly overestimate those in table 1 (by a few percent).

Table 1.

Numerical results obtained using the Rayleigh–Ritz and vortex lattice methods, described in § 3.3.2. Results are presented using a generalisation of (3.9); formula given in first row. This shows that the sheet’s aspect ratio, $H/L$ , exerts a weak effect on the ${\it\alpha}$ -coefficient, validating the accuracy of (3.9).

For completeness, we note that identical results are obtained using an approach analogous to that in § 3.2. That is, solving the eigenvalue problem for the unsteady thin plate equation (i.e. including the solid inertia term omitted in (3.10)) with a steady hydrodynamic load (specified by VLM for each $w_{n}(x)$ in the above described Rayleigh–Ritz method), as a function of ${\it\kappa}$ . As expected, bifurcation is found to occur at zero frequency, i.e. ${\it\Omega}=0$ (data not shown). Omission of solid inertia in (3.10) exactly simplifies the analysis by enabling the direct evaluation of ${\it\kappa}_{\mathit{lower}}$ using (3.11).

3.3.3 Effect of finite Reynolds number

Measurements on wings at high Reynolds number show that the lift coefficient and its aspect ratio correction both decrease by a factor of between 0.855 and 0.875 (Higgins Reference Higgins1929; Pelletier & Mueller Reference Pelletier and Mueller2000) for the Reynolds number range considered here, $\mathit{Re}\approx 10^{4}{-}10^{5}$ , and for aspect ratios $H/L\geqslant 1$ . This establishes that viscosity exerts a small effect on the overall lift, provided the aspect ratio exceeds unity. Combining this observation with (3.9) yields the required theoretical result:

Because the effects of viscosity have been shown to be small for aspect ratios greater than unity only, care must be taken when analysing measurements with $H/L<1$ . It is known that the sides of a wing generate vortices that can become trapped by the air flow and increase lift; the so-called ‘vortex-lift’ phenomenon (Anderson Reference Anderson1991). Since these trapped vortices occur at the sides of the wing, their effect on the overall lift is expected to increase with decreasing aspect ratio. These 3-D viscous effects are ignored in the 2-D inviscid calculations reported in the previous sections. Fully coupled Navier–Stokes-elastic simulations provide a rigorous approach for exploring such effects, but are hindered by the large Reynolds numbers used ( $\mathit{Re}\approx 10^{4}{-}10^{5}$ ). Numerical investigation provides an interesting avenue for further work, e.g. using the LES approach of Gilmanov et al. (Reference Gilmanov, Le and Sotiropoulos2015).

3.4.1 Effect of the added mass parameter ${\it\mu}$

Experiments by Kim et al. (Reference Kim, Cosse, Huertas Cerdeira and Gharib2013) show that the inverted flag loses its stability and deflects from its zero-deflection equilibrium at values of ${\it\kappa}_{\mathit{lower}}$ that are similar for two fluids: air and water. Note that the dimensionless parameters ${\it\kappa}$ and ${\it\mu}$ used here, which are taken from Kornecki et al. (Reference Kornecki, Dowell and O’Brien1976), are reciprocals of the ${\it\beta}$ and ${\it\mu}$ parameters of Kim et al. (Reference Kim, Cosse, Huertas Cerdeira and Gharib2013), respectively. We shall refer only to the definitions in (3.1) throughout this article.

The values of ${\it\mu}$ for air and water differ by three orders of magnitude, due to the difference in fluid density. For the $L=30$ cm inverted flag reported by Kim et al. (Reference Kim, Cosse, Huertas Cerdeira and Gharib2013), ${\it\mu}=0.38$ (air) and ${\it\mu}=320$ (water), representing strong fluid loading. The observed similitude with respect to ${\it\kappa}_{\mathit{lower}}$ , for varying ${\it\mu}$ , led to speculation of a divergence instability, though a definite conclusion could not be drawn based on only two data points. The theoretical analysis presented in the preceding section is consistent with these measurements and now establishes that a divergence instability indeed occurs for all fluids (at high Reynolds numbers). As such, the critical dimensionless flow speed, ${\it\kappa}_{\mathit{lower}}$ , is independent of the surrounding fluid and hence ${\it\mu}$ ; see (3.13).

3.4.2 Effect of the sheet’s aspect ratio, $H/L$

The new measurements in figure 3, for the critical flow speed at which the zero-deflection equilibrium bifurcates as a function of aspect ratio, are used to assess the validity and accuracy of (3.13). As the flow speed is increased from zero, the sheet remains undeflected initially while exhibiting very small-amplitude fluctuations; see figure 3. At higher flow speeds, flapping motion is initiated and increases in amplitude rapidly over a narrow range of flow speeds. The required bifurcation point (for comparison with theory) is selected by the flow speed at which the flapping amplitude is approximately 50 % of its maximum value, in this (narrow) flow speed range; this is obtained by measuring the flow speeds at which the amplitude is 25 % and 75 % of this maximum value, and taking their average. This allows the bifurcation point and its uncertainty to be quantified.

Figure 6 presents the results of this experimental analysis and a comparison to (3.13), for the aspect ratios measured; there are no adjustable parameters. Measurements in air on sheets of length 5.1 and 30 cm are used, dimensionless data from which are found to collapse precisely; see figure 6. This shows that the scaling parameters in (3.1) correctly capture the behaviour of this bifurcation phenomenon. We remind the reader that (3.13) holds for $H/L\geqslant 1$ only. Figure 6 shows that (3.13) correctly predicts the loss of stability of the zero-deflection equilibrium at high aspect ratio, with (slightly) decreasing accuracy as aspect ratio is reduced. The infinite aspect ratio result of ${\it\kappa}_{\mathit{lower}}=1.85$ strongly underestimates measurements of all the inverted flags studied here, which possess finite aspect ratios. This confirms the above theoretical consideration that the sides of the inverted flag reduce the hydrodynamic lift, requiring a higher flow speed for the sheet to become unstable. A general increase in ${\it\kappa}_{\mathit{lower}}$ with decreasing aspect ratio is predicted, and observed.

Figure 6.

Bifurcation point of the zero-deflection equilibrium of an inverted flag, showing the critical dimensionless flow speed, ${\it\kappa}_{\mathit{lower}}$ , as a function of aspect ratio, $H/L$ . Comparison of measurements in air (red dots) using sheets of length 5.1 cm and 30 cm, with the large aspect ratio ( $H/L\geqslant 1$ ) formula (3.13) (solid black line). The asymptotic limit $H/L\rightarrow \infty$ is indicated (dash-dotted black line). The (extrapolated) prediction of (3.13) is also shown for $H/L<1$ (dashed green line); this lies outside its range of validity and is given for reference only.

For reference, in figure 6 we also show the (extrapolated) prediction of (3.13) for $H/L<1$ , which is strictly outside of its regime of validity. Interestingly, measurements are found to strongly underestimate this theoretical prediction. This deviation is consistent with the vortex-lift phenomenon, discussed in § 3.3.3, whose effect will become more pronounced with decreasing aspect ratio. Its presence would enhance the lift experienced by the sheet, reducing the critical flow speed ${\it\kappa}_{\mathit{lower}}$ required for bifurcation, as observed. This possibility will be explored in a future study.

This completes our analysis of the zero-deflection equilibrium’s stability (at low flow speed). The measurements in figure 3 show that when this equilibrium bifurcates, large-amplitude flapping is launched immediately. In the next section, we explore the mechanisms giving rise to this behaviour.

4.1.1 Periodic vortex shedding from a bluff body

Periodic vortex shedding from bluff bodies has an extensive history of study (Mallock Reference Mallock1907; Bernard Reference Bernard1908; von Karman Reference von Karman1911, Reference von Karman1912; Fage & Johansen Reference Fage and Johansen1927; Schmitz Reference Schmitz1941; Roshko Reference Roshko1954; Bearman Reference Bearman1984; Knisely Reference Knisely1990), and the Strouhal number is used widely to characterise it. The Strouhal number corresponds physically to the ratio of a convective time scale for the impinging flow to the period of vortex shedding. It is (traditionally) defined

where $f_{\mathit{vs}}$ is the vortex shedding frequency (in Hz), $D$ is a characteristic length scale for the bluff body and $U$ the free-stream speed of the impinging flow. Measurements show that this vortex shedding is characterised by $\mathit{St}\approx 0.2$ for a wide range of bluff bodies in the Reynolds number range of interest here ( $\mathit{Re}\approx 10^{4}{-}10^{5}$ ), provided $D$ is chosen to be the dimension of the 2-D bluff body projected in the flow direction (Roshko Reference Roshko1954; Knisely Reference Knisely1990). This includes circular cylinders, wedges and thin plates at finite angles of attack. The latter case is most relevant to this study, because it closely mimics the geometry of a deflected inverted flag. This measured value of $\mathit{St}\approx 0.2$ is not unexpected because it indicates that the time scale for periodic vortex shedding is identical to that for convection of the free-stream flow over the body; the more natural use of angular frequency ${\it\omega}_{\mathit{vs}}=2{\rm\pi}f_{\mathit{vs}}$ leads to a modified Strouhal number $\mathit{St}^{\ast }\equiv {\it\omega}_{\mathit{vs}}D/U\approx 1$ , highlighting this balance of time scales.

Vortex shedding from a thin flat plate is certainly not expected for small angles of attack, where the flow remains attached and is predominately inviscid; see § 3.2. Indeed, thin airfoil theory is found to accurately predict the measured lift slope of a thin plate for angles of attack ${\lesssim}10^{\circ }$ (Pelletier & Mueller Reference Pelletier and Mueller2000). This lack of vortex shedding for angles of attack ${\lesssim}10^{\circ }$ is confirmed by the measurements of Knisely (Reference Knisely1990). Both Knisely (Reference Knisely1990) and Pelletier & Mueller (Reference Pelletier and Mueller2000) consider flows with Reynolds numbers $\mathit{Re}\approx 10^{4}{-}10^{5}$ , similar to the inverted flags here. Knisely (Reference Knisely1990) also shows that $\mathit{St}\approx 0.15$ for all angles of attack greater than ${\approx}10^{\circ }$ ; simulations verify these observations (Lam & Wei Reference Lam and Wei2010). We use these observed properties of a thin plate in our analysis of the flapping motion of an inverted flag in § 4.2.

4.1.2 Frequency lock-in during a VIV

Discussion in the previous section focuses on periodic vortex shedding from rigid bluff bodies. If the bluff body is elastic, the periodic hydrodynamic force applied by the same vortex shedding can induce a vibration of the body. A primary characteristic of the VIV of an elastic bluff body, is synchronisation of its motion with this periodic vortex shedding.

Measurements on a wide range of structures show that increasing the flow velocity from zero leads to distinct behaviour over a finite band of velocities (Brika & Laneville Reference Brika and Laneville1993; Williamson & Govardhan Reference Williamson and Govardhan2004; Gabbai & Benaroya Reference Gabbai and Benaroya2005), which we now summarise. This behaviour is again characterised by the Strouhal number, $\mathit{St}$ , which sets the natural frequency, $f_{\mathit{vs}}$ , for periodic vortex shedding; see § 4.1.1. Two cases are often considered in the literature: light and heavy fluid loading. In the context of (3.1), these cases correspond to ${\it\mu}\ll 1$ and ${\it\mu}\gtrsim O(1)$ , respectively; see note below for clarification of these regimes in existing literature measurements of VIVs.

Light fluid loading. Increasing the flow velocity from zero increases the vortex shedding frequency, $f_{\mathit{vs}}$ , at constant Strouhal number, typically $\mathit{St}\approx 0.2$ ; see § 4.1.1. When $f_{\mathit{vs}}$ is close to the natural resonant frequency of the elastic body, $f_{\mathit{res}}$ , the linear increase in $f_{\mathit{vs}}$ with flow speed ceases and the elastic body starts to vibrate at (or close to) its natural resonant frequency, with finite amplitude. Increasing the flow speed further does not alter this vibration frequency, with $f_{\mathit{vs}}\approx f_{\mathit{res}}$ , i.e. the vortex shedding frequency now departs from the $\mathit{St}\approx 0.2$ behaviour and $\mathit{St}$ decreases systematically. At higher flow speeds, where the Strouhal number is significantly lower than $\mathit{St}\approx 0.2$ , lock-in of $f_{\mathit{vs}}$ with $f_{\mathit{res}}$ is lost. The vortex shedding frequency returns to its previous behaviour, again satisfying $\mathit{St}\approx 0.2$ , and the oscillation amplitude drops dramatically.

Heavy fluid loading. The natural resonant frequencies of an elastic body in the presence and absence of fluid now differ considerably. For zero flow velocity, $f_{\mathit{res}}$ is significantly lower than its value when the fluid is absent, due to the strong inertial load exerted by the fluid that increases the body’s effective mass (Lindholm et al. Reference Lindholm, Kana, Chu and Abramson1965; Sader Reference Sader1998). Importantly, $f_{\mathit{res}}$ depends on the flow velocity and the body’s oscillation amplitude, which can both have a significant effect. As such, the lock-in observed for light fluid loading is modified with the vortex shedding frequency no longer being constant during a VIV (Gabbai & Benaroya Reference Gabbai and Benaroya2005).

The above mentioned Strouhal number trends are insensitive to Reynolds number for the range under consideration here, i.e. $\mathit{Re}\approx 10^{4}{-}10^{5}$ (Knisely Reference Knisely1990; Williamson & Govardhan Reference Williamson and Govardhan2004; Gabbai & Benaroya Reference Gabbai and Benaroya2005).

Note on existing literature measurements of VIVs. It is important to emphasise that measurements of heavy fluid loading in the literature typically refer to cases where ${\it\mu}\sim O(1)$ with ${\it\mu}<1$ (Williamson & Govardhan Reference Williamson and Govardhan2004; Gabbai & Benaroya Reference Gabbai and Benaroya2005). This is to be compared to the inverted flags considered here, that exhibit added mass parameters of ${\it\mu}=0.38$ for air and ${\it\mu}=320$ for water (Kim et al. Reference Kim, Cosse, Huertas Cerdeira and Gharib2013). These values of ${\it\mu}$ establish that the surrounding fluid significantly modifies the sheet’s resonant frequency in all measurements. Critically, measurements with ${\it\mu}>1$ have not been explored previously in the VIV literature, to the authors’ knowledge. The large values, ${\it\mu}\gg 1$ , for the inverted flag in water arise from the thin geometry of the sheet, whereas most VIV studies focus on circular cylinders (that naturally yield ${\it\mu}<1$ due to the difference in densities between solids and liquids), cf. added mass parameters of a circular cylinder and a flat rectangular sheet: $1/m^{\ast }$ in the Appendix of Williamson & Govardhan (Reference Williamson and Govardhan2004) and ${\it\mu}$ in (3.1), respectively. Note that the added mass parameter used in the literature is commonly $m^{\ast }$ (Williamson & Govardhan Reference Williamson and Govardhan2004), defining the ratio of inertia in the elastic body relative to that in the fluid, which is the reciprocal of ${\it\mu}$ (Kornecki et al. Reference Kornecki, Dowell and O’Brien1976).

4.1.3 Effect of damping on a VIV

Damping is known to strongly modify the characteristics of a VIV (Khalak & Williamson Reference Khalak and Williamson1999; Williamson & Govardhan Reference Williamson and Govardhan2004; Gabbai & Benaroya Reference Gabbai and Benaroya2005). Specifically, increasing damping in the system reduces the maximum amplitude of a VIV, and the flow velocity range over which it occurs. This property is illustrated in figure 6 of Gabbai & Benaroya (Reference Gabbai and Benaroya2005), where a comparison of measurements on elastic circular cylinders immersed in both air and water, is provided. The limit of high damping has been shown to reduce the amplitude of a VIV to zero, i.e. the VIV is eliminated (Khalak & Williamson Reference Khalak and Williamson1999).

Measurements of inverted flags are available in both air (figure 3) and water (Kim et al. Reference Kim, Cosse, Huertas Cerdeira and Gharib2013), allowing the effect of damping on the observed flapping to be assessed. In the next section, we shall also explore the effect of introducing additional damping to the inverted flag to see if it can eliminate flapping, in line with the above observation for a VIV (of an elastic cylinder).

Detailed measurements of the inverted flag’s dynamics are now presented to test for these signatures of a VIV, thus enabling an assessment of the hypothesis that flapping is a VIV.

4.2.1 Effect of additional damping

In § 3 we showed that the sheet’s zero-deflection equilibrium is unstable for normalised flow speeds greater than ${\it\kappa}_{\mathit{lower}}$ , because the sheet’s hydrodynamic lift (at small angle of attack) exceeds its elastic restoring force. This causes the sheet to deflect spontaneously from its zero-deflection equilibrium, leading to flow separation – which decreases the sheet’s lift slope and enhances its drag (Anderson Reference Anderson1991). This reduction in hydrodynamic lift limits the sheet’s deflection, which can in principle now reach a new finite-deflection equilibrium as the flow speed is increased immediately past ${\it\kappa}_{\mathit{lower}}$ . However, such an equilibrium is not observed in the measurements reported in figure 3 or by Kim et al. (Reference Kim, Cosse, Huertas Cerdeira and Gharib2013).

This prediction of a new (finite-deflection) equilibrium implicitly assumes that the hydrodynamic lift (and drag) are steady, which is not the case; periodic vortex shedding is always present, see § 4.1.1. As such, the above discussion of lift reduction refers to lift in a time-averaged sense. Importantly, the effects of such periodic forcing (superimposed on this average lift) can be reduced through the introduction of additional damping. This basic mechanism is highlighted in the literature, where damping has been shown to reduce the amplitude of a VIV and ultimately eliminate it; see § 4.1.3.

To test for the existence of this new equilibrium deflection, we introduce additional damping by lightly touching the inverted flag with a slender rigid pole; this takes kinetic energy away from the sheet when it oscillates. Figure 7 shows the results of these measurements over the complete range of air flow speeds studied, including the flapping region (highlighted). The pole is found to eliminate flapping and a new stable equilibrium is indeed observed in the flapping region; a restoring force is discovered when the pole is present, that drives the sheet to a previously unobserved stable finite-deflection equilibrium. The pole does not restrict motion of the sheet in both directions, but is applied on one side only – its effect is therefore to damp the motion rather than induce a stable equilibrium, which is illustrated in the following discussion.

Figure 7.

Measurements showing the effect of additional damping in the flapping region. New stable equilibrium in the flapping region (air flow speeds shown here: $3.5{-}5~\text{m}~\text{s}^{-1}$ ) is observed when a mechanically detached slender pole is introduced. At lower or higher flow speeds, the slender pole is not required because a stable equilibrium already exists, and introducing it has no effect. Asterisks indicate the range of motion observed in each measurement, which is predominantly due to twisting of the sheet; see movie in supplementary materials. This twisting motion may be due to the three-dimensional nature of the flow over the sheet. Sheet properties ( $L=30$  cm, $H=62$  cm).

A movie showing the effects of the pole is given in the supplementary materials available online at http://dx.doi.org/10.1017/jfm.2016.139 (the $3.5~\text{m}~\text{s}^{-1}$ image in figure 7 is taken from this movie). Strikingly, the sheet does not move from this new equilibrium once the pole is first removed, highlighting its inherent stability. Nonetheless, the sheet does start oscillating back and forth with increasing amplitude, and flapping motion is reinitiated in finite time. This shows that unsteady forces are responsible for the sheet’s instability – additional damping (induced by the pole) is required for the existence of the new stable equilibrium. Figure 7 shows that the magnitude of this new equilibrium deflection monotonically increases with flow speed and smoothly transitions into the (previously observed) deflected equilibrium above the flapping region (reported in figure 3). In this latter region, the pole has no effect on the equilibrium deflection. This measurement is in line with the above discussion regarding the effects of flow separation on the average lift, since increasing the flow speed will increase this lift and thus the sheet’s deflection. It also shows that flow unsteadiness causes flapping of the inverted flag.

The results in figure 7 are therefore in accordance with the known effect of damping on the elimination of a VIV (discussed in § 4.1.3).

4.2.2 Effect of changing the surrounding fluid

In the previous section, we showed that strongly increasing the damping by external means (using a slender pole) stops the inverted flag from flapping and gives rise to a new stable equilibrium. If flapping is indeed a VIV, then changing the fluid from air to water is again expected to impact its dynamics because this also changes the damping.

Measurements in water by Kim et al. (Reference Kim, Cosse, Huertas Cerdeira and Gharib2013) are used for this purpose and show that the inverted flag’s flapping dynamics are modified significantly relative to air. Namely, the onset of flapping at low flow speed, i.e. ${\it\kappa}_{\mathit{lower}}$ , is similar to air (as noted in § 3.4.1), but the speed ${\it\kappa}_{\mathit{upper}}$ at which flapping ceases abruptly and a stable large-deflection equilibrium emerges, decreases by a factor of two. As such, the flow speed range over which flapping is observed is smaller in water, in comparison to air. Moreover, measurements in water show that as the flow speed increases, the onset of flapping at ${\it\kappa}_{\mathit{lower}}$ occurs at a greatly reduced amplitude relative to operation in air. The amplitude is enhanced with increasing flow speed, but is smaller than measurements in air, cf. figures 2 and 4 of Kim et al. (Reference Kim, Cosse, Huertas Cerdeira and Gharib2013); note that an inverse flow speed, ${\it\beta}\equiv 1/{\it\kappa}$ , is used in that study. Operation in water is found to restrict the motion in many cases, particularly at low flow speeds, with the sheet not crossing its zero-deflection position during an oscillation, i.e. flapping only occurs on one side with reduced amplitude (Kim et al. Reference Kim, Cosse, Huertas Cerdeira and Gharib2013).

These observations are similar to known properties of a VIV, where an increase in damping reduces (i) the oscillation amplitude of a VIV, and (ii) the flow speed range over which it occurs; see § 4.1.3.

4.2.3 Synchronisation of flapping motion with vortex shedding

A key attribute of a VIV is the synchronisation of vortex shedding from the elastic body with the body’s motion. Kim et al. (Reference Kim, Cosse, Huertas Cerdeira and Gharib2013) provided flow visualisation measurements of an inverted flag flapping in water. These measurements are reproduced in figure 8, showing the motion of the sheet over a half-period of the flapping oscillation. These measurements clearly demonstrate that the formation and subsequent shedding of vortices from the leading (and trailing) edges of the sheet are synchronised with the sheet’s motion, as required. This experimental observation is also confirmed by recent computational simulations of the inverted flag, using an LES approach to capture the high Reynolds numbers involved (Gilmanov et al. Reference Gilmanov, Le and Sotiropoulos2015).

Figure 8.

Sequential vortex formation process from an inverted flag undergoing flapping in water. Non-dimensional vorticity contours are indicated: $\overline{{\it\omega}}={\it\omega}_{\mathit{vs}}L/U$ , where ${\it\omega}_{\mathit{vs}}$ is the vorticity (not to be confused with the angular frequency, ${\it\omega}$ ). Measurements taken from figure 6(a) of Kim et al. (Reference Kim, Cosse, Huertas Cerdeira and Gharib2013).

4.2.4 Strouhal number of flapping

Vortex-induced vibrations are initiated typically at a Strouhal number of $\mathit{St}\approx 0.2$ . Here, we explore the dynamics of the inverted flag in the flapping region to see if it also exhibits this behaviour. Example measurements are shown for an inverted flag with $L=5.1$ cm and $H=6.4$ cm, while noting that similar behaviour is observed for other aspect ratios and length.

The flapping frequency is obtained from stroboscopic images of the sheet, which are digitised to yield time series of its displacement. Deflections of the sheet’s free end in both the $x$ - and $y$ -directions are measured; see figure 2 for coordinate system. Figure 9 presents a sample of these stroboscopic images and their corresponding time series (for the $y$ -component only; the $x$ -component exhibits similar behaviour), as a function of ${\it\kappa}$ . Data at low flow speed in the flapping region clearly show that the sheet’s motion is highly periodic; this data is discussed further in § 4.2.7. The time series data is Fourier transformed to yield the required flapping frequency, $f_{\mathit{res}}$ , which is selected to be the spectral component of largest magnitude. These measured values for $f_{\mathit{res}}$ are shown in figure 9, for three values of ${\it\kappa}$ within the flapping region.

Figure 9.

Stroboscopic images (a–e) and corresponding times series (f–j) of the sheet’s motion (directly below), as a function of ${\it\kappa}$ . The $y$ -coordinate listed is the position of the sheet’s free end; see figure 2. The flapping frequency as a function of ${\it\kappa}$ is also indicated. Sheet with $L=5.1~\text{cm}$ and $H=6.4~\text{cm}$ , measured in air.

Figure 10(a) gives the complete dataset for $f_{\mathit{res}}$ , over all values of ${\it\kappa}$ measured in the flapping region, i.e. for ${\it\kappa}>{\it\kappa}_{\mathit{lower}}$ . This shows that the flapping frequency, $f_{\mathit{res}}$ , varies slightly with flow speed, but is significantly lower than the natural resonant frequency of the sheet in a quiescent fluid. As discussed in § 4.1.2, the resonant frequency of an elastic structure under heavy fluid loading depends on (i) the added mass parameter, ${\it\mu}$ , (ii) the flow speed and (iii) its oscillation amplitude. The observations in figure 10(a) are therefore in line with this expectation, because the fluid loading is strong ( ${\it\mu}=0.38$ ). Moreover, we observe that the maximum flapping frequency, $f_{\mathit{res}}\approx 8~\text{Hz}$ at ${\it\kappa}\approx 6$ , coincides with the sheet exhibiting a maximum deflection oriented $90^{\circ }$ to the flow direction – illustrating the connection between the flapping amplitude (and shape of the vibrating mode) with $f_{\mathit{res}}$ .

Figure 10.

Flapping dynamics of the inverted flag studied in figure 9. (a) Flapping frequency, $f_{\mathit{res}}$ , and (b) Strouhal number, $St$ .

Corresponding results for the Strouhal number of the inverted flag are provided in figure 10(b). In accordance with (4.1), the Strouhal number for an inverted flag is defined (Kim et al. Reference Kim, Cosse, Huertas Cerdeira and Gharib2013)

where $f_{\mathit{res}}$ is the flapping (resonant) frequency and $A$ is the maximum amplitude of flapping (peak-to-peak in the $y$ -direction; see figure 2) to account for the deformable geometry of the flapping sheet and allow for comparison to previous studies on other bodies. Strikingly, the data in figure 10(b) shows that flapping is initiated at $\mathit{St}\approx 0.15$ , with the Strouhal number decreasing systematically as the flow speed increases – flapping ceases when the Strouhal number is below half its initial value.

These values for $St$ are quantitatively similar to those reported previously (Kim et al. Reference Kim, Cosse, Huertas Cerdeira and Gharib2013) for inverted flags made from the (larger) sheet of length $L=30$ cm. The sheet length-to-thickness ratio is identical for both inverted-flag sizes, $L=5.1$ and 30 cm, and therefore, the flow speed where flapping begins is identical in both cases; see (3.1). As such, the Reynolds number for the $L=30$ cm flag studied by Kim et al. (Reference Kim, Cosse, Huertas Cerdeira and Gharib2013) is six times larger than for the $L=5.1$ cm flag reported in figures 9 and 10, while their dynamics occur at the same ${\it\kappa}$ and ${\it\mu}$ values. The $L=5.1$ and 30 cm flags therefore probe different flow regimes. This quantitative behaviour of the Strouhal number with increasing flow speed, and its insensitivity to Reynolds number, are identical to observations of VIVs of a broad range of mechanical structures; see § 4.1.2. The low Reynolds number computational simulations of inverted flag flapping by Ryu et al. (Reference Ryu, Park, Kim and Sung2015) exhibit similar behaviour, even though the Reynolds numbers are two orders of magnitude smaller.

Flapping in water (figure 4a of Kim et al. (Reference Kim, Cosse, Huertas Cerdeira and Gharib2013)) is somewhat different to air because the sheet does not always exhibit (the same qualitative) side to side oscillations, but instead flaps on one side predominantly – this modified mode shape leads to a reduction in amplitude and hence Strouhal number; see § 4.2.2.

4.2.5 Can a VIV be launched at ${\it\kappa}_{\mathit{lower}}$ ?

Next, we examine whether it is possible for a VIV to be launched immediately following the divergence instability at ${\it\kappa}_{\mathit{lower}}$ . This is performed using a scaling analysis of the motion under the stated hypothesis that flapping is a VIV. This is motivated by the experimental observation that flapping starts immediately at ${\it\kappa}_{\mathit{lower}}$ ; see figure 3.

The resonant frequency of a 2-D cantilevered sheet, in the absence of fluid, is (Looker & Sader Reference Looker and Sader2008)

where $C_{1}=1.8751$ (for the fundamental flexural mode). Immersion in fluid increases the effective mass of the sheet, due to the fluid’s inertial load, and thus lowers the sheet’s resonant frequency. This increase in mass is accounted for by replacing the areal mass density ${\it\rho}_{s}h$ with ${\it\rho}_{s}h(1+a{\it\mu})$ (Lindholm et al. Reference Lindholm, Kana, Chu and Abramson1965), where $a$ is an order one coefficient, which for simplicity we approximate by $a\approx 1$ . This gives the required expression for the resonant frequency of the sheet in fluid:

Substituting (4.4) into (4.2), and making use of (3.1), then gives

defining the connection between the Strouhal number for flapping, $St$ , and the dimensionless flow speed, ${\it\kappa}$ . Equation (4.5) predicts $\mathit{St}\sim 1/\sqrt{{\it\kappa}}$ for constant $A/L$ and ${\it\mu}$ . This decreasing behaviour in $St$ is observed in figure 10(b) for larger values of ${\it\kappa}$ , where $A/L$ is approximately constant. We refrain from further direct comparison with (4.5), because the resonant frequency, $f_{\mathit{res}}$ , and hence the influence of the added mass parameter, ${\it\mu}$ , depend on ${\it\kappa}$ (see figure 10 a), an effect that has not been calculated.

We now specify the minimum and maximum values for the flapping amplitude, $A$ . The minimum limit for $A$ is set by the requirement that periodic vortex shedding must exist for a VIV to be launched. As discussed in § 4.1.1, the sheet must present a finite deflection that exceeds a minimum angle of attack for flow separation and periodic vortex shedding to occur. Using the measured value for a flat plate of ${\approx}10^{\circ }$ (Knisely Reference Knisely1990), as an approximation, gives the required lower limit for the flapping amplitude:

Existence of a minimum amplitude is clearly observed in the inverted-flag measurements reported in figure 3, and (4.6) agrees with minimum values for the flapping amplitude in figures 2 and 4 of Kim et al. (Reference Kim, Cosse, Huertas Cerdeira and Gharib2013).

The maximum flapping amplitude of an inverted flag is limited by its geometry: amplitudes greater than twice the sheet length are not physically realisable, i.e.

Substituting (4.6) and (4.7) into (4.5) gives

Finally, we note that VIVs occur over the approximate range $\mathit{St}\in (0.1,0.2)$ for a wide range of bluff bodies; see § 4.1.2. Substituting these approximate upper and lower limits for $St$ into (4.8), gives the required inequality

which specifies the range of dimensionless flow speeds, ${\it\kappa}$ , for which a VIV can exist. Values of ${\it\kappa}$ outside this range require amplitudes that are either not possible geometrically (above the upper limit of this inequality), or do not allow the inverted flag to exhibit periodic vortex shedding (below the lower limit).

The added mass parameter for the inverted flags studied in figure 3 (in air) is ${\it\mu}=0.38$ , and thus (4.9) becomes

Comparing this inequality with (3.13) (and the data in figure 6) shows that ${\it\kappa}_{\mathit{lower}}$ lies within the required flow speed range where a VIV can exist. This finding also holds true for inverted flags immersed in water, corresponding to ${\it\mu}\gg 1$ . We therefore conclude that a VIV can indeed be launched at ${\it\kappa}_{\mathit{lower}}$ , for the measurements presented.

Light fluid loading. Importantly, (3.13) does not satisfy (4.9) for light fluid loading, i.e. ${\it\mu}\ll 1$ , showing that an inverted flag cannot undergo a VIV in such cases: (i) ${\it\kappa}_{\mathit{lower}}>1.85$ , (ii) ${\it\kappa}_{\mathit{lower}}$ is independent of ${\it\mu}$ , and (iii) both the upper and lower limits of (4.9) approach zero as ${\it\mu}\rightarrow 0$ . The critical flow speed, ${\it\kappa}_{\mathit{lower}}$ , therefore lies outside of the required range for a VIV in (4.9) and an allowable oscillation amplitude does not exist, for very small ${\it\mu}$ . Physically, this is because the resonant frequency of the inverted flag, $f_{\mathit{res}}$ , under light fluid loading is significantly lower than the vortex shedding frequency, $f_{\mathit{vs}}$ , at ${\it\kappa}_{\mathit{lower}}$ – increasing the flow speed above ${\it\kappa}_{\mathit{lower}}$ , enhances $f_{\mathit{vs}}$ and thus eliminates the possibility of lock-in at any flow speed. Note that $f_{\mathit{res}}$ is independent of flow speed for ${\it\mu}\ll 1$ .

4.2.6 Is flapping a vortex-induced vibration?

We now summarise the above observations and analyses of an inverted flag’s dynamics:

1. (1) Flapping occurs for Strouhal numbers over the approximate range $\mathit{St}\in (0.1,0.2)$ . It is initiated at a Strouhal number, $\mathit{St}\approx 0.2$ , which decreases systematically with increasing flow speed. Flapping ceases when the Strouhal number is less than approximately half of its initial value.

2. (2) The Strouhal number dependence on flow speed is insensitive to Reynolds number, as also observed for VIVs (see § 4.1.2). For the inverted flags studied, it does not change when the Reynolds number is varied by a factor of six.

3. (3) Periodic vortex shedding from the inverted flag always occurs after its divergence instability – due to the inverted flag presenting its sharp leading edge at large angle of attack to the impinging flow.

4. (4) Flapping is observed to occur above a minimum amplitude (see Kim et al. (Reference Kim, Cosse, Huertas Cerdeira and Gharib2013) and figure 3), as predicted by (4.6) – this is the regime where flow separation and periodic vortex shedding exist (Knisely Reference Knisely1990; Pelletier & Mueller Reference Pelletier and Mueller2000).

5. (5) The inverted flags studied here and by Kim et al. (Reference Kim, Cosse, Huertas Cerdeira and Gharib2013) operate under heavy fluid loading, i.e. ${\it\mu}\gtrsim O(1)$ . The flapping frequency depends on the surrounding fluid, varying with the flow speed and the flapping amplitude – as expected for the resonant frequency of elastic structures under heavy fluid loading.

6. (6) Changing the damping experienced by the inverted flag modifies its dynamics, also in accordance with what is expected for a VIV:

   1. (i) The amplitude of flapping and the flow speed range over which it occurs are reduced as damping increases. Namely, (a) the value of ${\it\kappa}_{\mathit{lower}}$ where flapping starts at low flow speed is unchanged when air or water is used, and (b) the upper value of ${\it\kappa}$ where flapping ceases is reduced by a factor of two in water, relative to air, and the flapping amplitude is smaller.

   2. (ii) Application of strong damping, to suppress the effects of unsteady hydrodynamic forces, eliminates flapping and leads to emergence of a new stable finite-deflection equilibrium. Flapping re-emerges once this damping is removed, showing that unsteady fluid forces are responsible for flapping.

7. (7) A scaling analysis shows that a VIV can be immediately launched following the divergence instability at ${\it\kappa}_{\mathit{lower}}$ , provided the inverted flag operates under heavy fluid loading; all inverted flags studied here satisfy this criterion. Operation under light fluid loading suppresses flapping.

8. (8) Periodic vortex shedding from the leading and trailing edges of the inverted flag is synchronised with its flapping motion.

VIVs of elastic bluff bodies reported in the literature exhibit the same above-mentioned characteristics (for flapping of an inverted flag). This multifaceted connection therefore provides considerable evidence to support the hypothesis that flapping of an inverted flag is indeed a VIV. It is initiated by periodic vortex shedding from the sharp leading edge of the inverted flag at large angle of attack – which occurs when the inverted flag exhibits a divergence instability and deflects.

4.2.7 Transition to chaos

Simulations and measurements of conventional flags show that their dynamics can exhibit a transition to chaos as the sheet’s bending rigidity is reduced, or equivalently the flow speed is increased (Connell & Yue Reference Connell and Yue2007; Alben & Shelley Reference Alben and Shelley2008; Michelin et al. Reference Michelin, Llewellyn Smith and Glover2008; Ait Abderrahmane et al. Reference Ait Abderrahmane, Paidoussis, Fayed and Ng2012; Chen et al. Reference Chen, Jia, Wu, Yin and Ma2014). In this section, we examine the possibility that chaos also occurs in the flapping region of an inverted flag.

The measurements in figure 9(f–j) show that flapping initially undergoes periodic motion following the divergence instability, at ${\it\kappa}_{\mathit{lower}}$ . This periodic motion is maintained over a broad range of flow speeds, $4.25<{\it\kappa}<11.4$ . For higher speeds, $11.4<{\it\kappa}<13.4$ , which are close to the flow speed where flapping ceases abruptly, aperiodic motion emerges and intensifies with increasing flow speed. Motivated by observations on conventional flags (see above), we now explore the possibility that this aperiodic motion is chaotic.

While there are many definitions for chaos, one dominant property is that the motion must exhibit a sensitive dependence on its initial conditions and be bounded (Strogatz Reference Strogatz1994). We therefore study the observed aperiodic motion in detail and compare extracts of its time series that have nearly identical initial trajectories. Figure 11 shows a sample comparison for the $y$ -coordinate of the sheet’s free end, obtained by shifting and overlaying different parts of the time series; see figure 11(b) for overlay. Strikingly, while these time series extracts (red and green curves) have very similar trajectories at small time (indicated), they deviate abruptly in finite time and exhibit wildly different behaviour at longer times. Identical behaviour is found for the corresponding $x$ -component, which also overlay at small time and deviate at precisely the same time as the $y$ -component (results not shown). This demonstrates the existence of bounded motion of the sheet with a sensitive dependence on initial conditions – establishing the existence of chaos.

Figure 11.

Demonstration of chaos in the flapping dynamics of the inverted flag studied in figure 9, at ${\it\kappa}=13.36$ . Data taken from figure 9(i). The highlighted green (left) and red (right) portions of (a) above are time shifted and overlaid in (b). While these time shifted data precisely overlap for small time (indicated), an abrupt deviation is observed in finite time demonstrating a sensitive dependence on initial conditions. Identical behaviour is observed for the corresponding $x$ -coordinate of the sheet’s free end (data not shown).

The inverted flag therefore undergoes chaotic dynamics, over a small flow speed range, immediately prior to the abrupt emergence of a stable deflected equilibrium at high flow speed. We also note that the frequency spectrum of the sheet’s motion in this chaotic regime is dense (not shown), while the frequency component with the largest amplitude decreases monotonically with increasing flow speed; see figure 10(a). These results are to be compared to the known behaviour of conventional flags, where chaos is theoretically predicted to emerge at ${\it\mu}\approx 8$ upon a reduction in the added mass parameter, ${\it\mu}$ , from higher values; see Connell & Yue (Reference Connell and Yue2007), while noting that the definition for ${\it\mu}$ in that study is the reciprocal of that used here, cf. (1.1) of Connell & Yue (Reference Connell and Yue2007) with (3.1). The inverted flag studied in figure 11 has a value for the added mass parameter of ${\it\mu}=0.38$ , and thus falls within the above mentioned range, ${\it\mu}\lesssim 8$ , for which a conventional flag also exhibits chaos.

The properties of the inverted flag’s chaotic motion, and its relation to chaos observed for a conventional flag, presents an interesting area for further investigation.