2.1 No petioles

We first consider inverted flags without petioles, i.e. the trapezoidal sheet is attached directly to the clamp, formed from a single sheet of polycarbonate. The free stream impinges on the leading free edge of the sheet while the trailing edge of the sheet is clamped between two thin but rigid steel strips. The heights of all trapezoidal sheets are fixed at $H=19.05$  cm and their lengths are $L=15.24$  cm (giving $H/L=1.25$ ), which results in a Reynolds number of $Re\sim O(10^{4})$ . We tested polycarbonate sheets with thicknesses of $h=0.25$ and 0.38 mm, corresponding to mass ratios of $\unicode[STIX]{x1D707}=0.61$ and 0.40, respectively. The trapezoidal shape of the sheets is adjusted in the range $0\leqslant \unicode[STIX]{x1D6E5}/H\leqslant 1.75$ for sheets with $\unicode[STIX]{x1D707}=0.40$ , and $0\leqslant \unicode[STIX]{x1D6E5}/H\leqslant 1$ for sheets with $\unicode[STIX]{x1D707}=0.61$ . Note that $\unicode[STIX]{x1D6E5}/H=0$ corresponds to a triangular morphology.

Figure 3.

Schematic of an inverted flag of rectangular morphology with a petiole. (a) Side view, and (b) top/camera view with straight rest position and flapping dynamics overlaid. The shaded area corresponds to the elastic polycarbonate sheet with a small tab-like overlap region used to join it to the petiole.

2.2 Petioles

The effect of including a petiole is studied for sheets of rectangular planform only. The elastic sheet and petiole are both fabricated from polycarbonate sheets. Each petiole is constructed by sandwiching two slender polycarbonate strips and bonding them with cyanoacrylate adhesive. The inverted flag is constructed from a single large rectangular sheet that is joined to the petiole using a small rectangular tab-like overlap region on the sheet; see figure 3(a). The resulting inverted flag is mounted in the wind tunnel by rigidly clamping the downstream end of the petiole to the steel strips (which are described in § 2.1) with the flow impinging on the sheet’s free edge. The petiole length $L_{p}$ , height $H_{p}$ and thickness $h_{p}$ are illustrated in figures 3(a) and 3(b). The petiole length, $L_{p}$ , is varied relative to sheet length, $(L-L_{p})$ , while fixing total length of the inverted flag, $L$ , height $H$ and the sheet thickness $h$ . The petiole’s flexural rigidity is adjusted by changing its thickness, $h_{p}$ . Parameter values used in this study are given in table 1. By approximating the petiole as a single uniform strip of polycarbonate, the rigidity ratio between the sheet and the petiole is $(h/h_{p})^{3}$ ; see (1.3).

Also shown in table 1 are measured values for the inverted flags’ maximum flapping frequencies (driven by the impinging flow) and their natural resonant frequencies (in the absence of flow). The flapping frequency is determined via the average number of cycles per unit time, because the dynamics is highly periodic, and $f_{max}$ is the maximum frequency of the flapping dynamics as the flow speed is varied (excluding chaotic flapping).

Table 1.

Dimensions, petiole description and measured frequencies of rectangular inverted flags with a petiole; Flag E is a rectangular inverted flag with no petiole that is used for reference. These flags are studied in § 5. Here, $f_{max}$ is the measured flapping frequency with an impinging flow and $f_{res}$ is its measured natural frequency in the absence of flow. All length/height/thickness values are reported in cm and frequencies are in Hz.

4.1 Strip model for arbitrary planform $(AR\rightarrow \infty )$

In the limit, $AR\rightarrow \infty$ , as defined in (1.4), the pressure distribution along the cantilevered sheet can be rigorously calculated by dividing the sheet into infinitesimal rectangular strips in the direction of flow (at each spanwise location). This limit corresponds to a sheet that is infinitely wide relative to its length. The dimensionless pressure jump distribution (scaled by $\unicode[STIX]{x1D70C}U^{2}A_{disp}/L$ , where $A_{disp}$ is the magnitude of the sheet’s free end static displacement) across each strip is then given by that of an infinitely wide (2-D) rectangular cantilevered sheet (Kornecki et al. Reference Kornecki, Dowell and O’Brien1976),

(4.1) $$\begin{eqnarray}\displaystyle p(x,z)={\displaystyle \frac{g(x,z)-g(x_{1}(z),z)}{\sin \unicode[STIX]{x1D703}(x,z)}}, & & \displaystyle\end{eqnarray}$$

where

(4.2a ) $$\begin{eqnarray}\displaystyle & \displaystyle g(x,z)=-{\displaystyle \frac{2}{\unicode[STIX]{x03C0}}}\int _{0}^{\unicode[STIX]{x03C0}}{\displaystyle \frac{\sin ^{2}\unicode[STIX]{x1D719}}{\cos \unicode[STIX]{x1D719}-\cos \unicode[STIX]{x1D703}(x,z)}}{\displaystyle \frac{\unicode[STIX]{x2202}w(\unicode[STIX]{x1D701},z)}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}\,\text{d}\unicode[STIX]{x1D719}, & \displaystyle\end{eqnarray}$$

(4.2b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}=x_{0}(z)+{\displaystyle \frac{x_{1}(z)-x_{0}(z)}{2}}(1+\cos \unicode[STIX]{x1D719}), & \displaystyle\end{eqnarray}$$

(4.2c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}(x,z)=\cos ^{-1}\left[{\displaystyle \frac{2x-x_{1}(z)-x_{0}(z)}{x_{1}(z)-x_{0}(z)}}\right]. & \displaystyle\end{eqnarray}$$

$w(x,z)$

$A_{disp}$

$x$

$y$

$L$

$x_{0}(z)$

$x_{1}(z)$

$x$

$z$

$\unicode[STIX]{x1D703},\unicode[STIX]{x1D719}\in [0,\unicode[STIX]{x03C0}]$

$z$

$x$

$\unicode[STIX]{x1D703}$

$\unicode[STIX]{x1D701}$

$\unicode[STIX]{x1D719}$

To determine the dimensionless flow speed, $\unicode[STIX]{x1D705}_{lower}$ , at which divergence occurs we make use of the small-deflection theory for thin plates (Landau & Lifshitz Reference Landau and Lifshitz1970),

(4.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}^{4}w(x,z)=\unicode[STIX]{x1D705}\,p(x,z). & & \displaystyle\end{eqnarray}$$

Equation (4.3) is then multiplied by $w(x,z)$ and integrated over the sheet’s surface, $S$ , while imposing the sheet’s boundary conditions and noting that $w(x,z)$ varies slowly with respect to $z$ (when $H/L\gg 1$ ), to give

(4.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{lower}^{\infty }={\displaystyle \frac{\displaystyle \int _{S}\left(\displaystyle \frac{\unicode[STIX]{x2202}^{2}w}{\unicode[STIX]{x2202}x^{2}}\right)^{2}\,\text{d}S}{\displaystyle \int _{S}p\,w\,\text{d}S}}, & & \displaystyle\end{eqnarray}$$

where the superscript, $\infty$ , denotes use of the current strip model and is valid for $AR\rightarrow \infty$ . Since the pressure, $p$ , is linearly dependent on the displacement, $w$ , it then follows that (4.4) is independent of the displacement magnitude, $A_{disp}$ , as required.

In Sader et al. (Reference Sader, Cossé, Kim, Fan and Gharib2016a ), the deflection function $w$ is determined using a Rayleigh–Ritz approach where $w$ is expressed as a general power series in $x$ with unknown coefficients; these unknown coefficients are then determined from the stationary point of (4.4). To simplify implementation here, we instead make use of the known deflection function for a rectangular cantilevered sheet ( $\unicode[STIX]{x1D6E5}/H=1$ ) under a static point load at its free end,

(4.5) $$\begin{eqnarray}\displaystyle w(x,z)={\textstyle \frac{1}{2}}(2+x)(1-x)^{2}, & & \displaystyle\end{eqnarray}$$

which enforces the free end condition at $x=0$ and the clamp at $x=1$ . This facilitates analytical solution of $\unicode[STIX]{x1D705}_{lower}$ for sheets of arbitrary planform by substituting (4.1) and (4.5) into (4.4). Because (4.5) is an approximation to the true deflection function of a sheet of arbitrary morphology, this approach guarantees that the resulting value for $\unicode[STIX]{x1D705}_{lower}^{\infty }$ is strictly an upper bound (Leissa Reference Leissa1969). This is consistent with the neglect of vortex lift, which leads to an overestimate of $\unicode[STIX]{x1D705}_{lower}$ (see above), ensuring the presented theory produces a strict upper bound for $\unicode[STIX]{x1D705}_{lower}$ . It is important to note that instead of (4.5), a general power series expansion could be used, as in Sader et al. (Reference Sader, Cossé, Kim, Fan and Gharib2016a ), at the expense of increased complexity. Clearly, care must be exercised when the expected deflection mode shape deviates significantly from (4.5), which will be discussed in § 4.2.1.

Substituting (4.5) into (4.1) gives the required pressure distribution,

(4.6) $$\begin{eqnarray}\displaystyle p(x,z)=-{\displaystyle \frac{3(x_{1}-x_{0})\sin \unicode[STIX]{x1D703}}{8(1+\cos \unicode[STIX]{x1D703})}}\left[{\displaystyle \frac{8(x_{1}^{2}-1)}{x_{1}-x_{0}}}+2(x_{0}+3x_{1})\cos \unicode[STIX]{x1D703}+(x_{1}-x_{0})\cos 2\unicode[STIX]{x1D703}\right], & & \displaystyle\end{eqnarray}$$

where $x_{0}$ , $x_{1}$ and $\unicode[STIX]{x1D703}$ are defined in (4.2c ) and figure 5, and depend on $x$ and $z$ . Equations (4.4)–(4.6) enable the divergence instability of an inverted cantilever sheet of arbitrary planform to be calculated directly and trivially.

4.2 Trapezoidal planform

The general strip theory in the previous section is now used to calculate $\unicode[STIX]{x1D705}_{lower}^{\infty }$ for an inverted flag of trapezoidal morphology; this is corrected for finite $AR$ in § 4.3. Making use of (4.5) and (4.6), the integration of (4.4) over the trapezoidal planform then yields

(4.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{lower}^{\infty }=\left\{\begin{array}{@{}ll@{}}\displaystyle {\displaystyle \frac{768}{\unicode[STIX]{x03C0}}}{\displaystyle \frac{3+{\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}}{121+404{\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}}},\quad & 0\leqslant {\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}<1,\\[24.0pt] \displaystyle {\displaystyle \frac{1792}{\unicode[STIX]{x03C0}}}{\displaystyle \frac{3+{\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}}{186+1039{\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}}},\quad & 1\leqslant {\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}\lesssim O(10),\end{array}\right. & & \displaystyle\end{eqnarray}$$

where the origin of the $O(10)$ upper limit is explained below in § 4.2.1. Note that (4.7) gives $\unicode[STIX]{x1D705}_{lower}^{\infty }=1.86$ for an inverted flag of rectangular morphology, i.e. $\unicode[STIX]{x1D6E5}/H=1$ , whereas the full power series approach of Sader et al. (Reference Sader, Cossé, Kim, Fan and Gharib2016a ) gives $\unicode[STIX]{x1D705}_{lower}^{\infty }=1.85$ , which is in excellent agreement.

Figure 6(a) provides numerical results for $\unicode[STIX]{x1D705}_{lower}^{\infty }$ as a function of the trapezoidal shape parameter, $\unicode[STIX]{x1D6E5}/H$ , showing that the flow speed at which divergence occurs increases with decreasing $\unicode[STIX]{x1D6E5}/H$ . For $\unicode[STIX]{x1D6E5}/H=0$ , corresponding to a triangular sheet whose leading edge is a point, equation (4.7) gives $\unicode[STIX]{x1D705}_{lower}^{\infty }=6.06$ . In the opposite limit, $\unicode[STIX]{x1D6E5}/H\rightarrow \infty$ , which is an inverted flag with a very wide leading edge relative to its clamped end – i.e. a reversed triangle – (4.7) predicts $\unicode[STIX]{x1D705}_{lower}^{\infty }=0.549$ . However, the mode shape in (4.5) obviously does not hold when $\unicode[STIX]{x1D6E5}/H\gg 1$ , since the flag’s rigidity at the clamp will approach zero relative to its leading edge.

In the following subsection, we therefore examine the effect of varying the deflection function (4.5) on the calculated $\unicode[STIX]{x1D705}_{lower}^{\infty }$ . This is performed for the limiting cases of small and large $\unicode[STIX]{x1D6E5}/H$ .

Figure 6.

Dimensionless flow speeds, $\unicode[STIX]{x1D705}_{lower}^{\infty }$ and $\unicode[STIX]{x1D705}_{lower}$ , for the divergence instability of a trapezoidal inverted flag; superscript $\infty$ indicates infinite aspect ratio $AR$ . (a) Theory for the divergence instability of a trapezoidal inverted flag of infinite $AR$ . Dimensionless flow speed, $\unicode[STIX]{x1D705}_{lower}^{\infty }$ , as a function of the shape parameter, $\unicode[STIX]{x1D6E5}/H$ , for an inverted flag of trapezoidal morphology (no petiole). Equation (4.7) holds for $\unicode[STIX]{x1D6E5}/H\lesssim O(10)$ , while (4.8) is derived for $\unicode[STIX]{x1D6E5}/H\gg 1$ . This theory is valid for $H/L\rightarrow \infty$ , while measurements are performed on sheets of finite $H/L=O(1)$ . (b) Finite aspect ratio correction for the dimensionless flow speed of the trapezoidal sheets studied in § 3. Comparison of (4.10), which makes use of the generalised aspect ratio in (4.11), to numerical results obtained with the vortex lattice method (VLM).

4.2.1 Varying the deflection function

$\unicode[STIX]{x1D6E5}/H=0$ corresponds to a triangular sheet whose leading edge is a point. Moreover, (4.5) is derived for a rectangular cantilevered sheet under a static point load at its free end. We therefore vary the mode shape by instead using the corresponding deflection for a triangular cantilever sheet, which gives $w(x,z)=(1-x)^{2}$ . Calculating the hydrodynamic pressure from (4.1), using this new mode shape, and substituting the result into (4.4), gives $\unicode[STIX]{x1D705}_{lower}^{\infty }=64/(3\unicode[STIX]{x03C0})=6.79$ ; this is to be compared to $\unicode[STIX]{x1D705}_{lower}^{\infty }=6.06$ , obtained using (4.7). Since the Rayleigh quotient in (4.4) guarantees an upper bound (Leissa Reference Leissa1969), this analysis establishes that $\unicode[STIX]{x1D705}_{lower}^{\infty }=6.06$ is more accurate.

The value $\unicode[STIX]{x1D6E5}/H\gg 1$ is an inverted flag with a very wide leading edge relative to its (trailing) clamped end, i.e. a reversed triangle. Here, only a very small strip of elastic material joins the sheet to the clamp, relative to the large lift surface. The inverted flag will therefore deflect in a linear (and rigid body) fashion away from the clamp, with a small ‘boundary layer’ near the clamp that bends elastically. Any non-zero flow speed will deflect the flag in this limit, giving $\unicode[STIX]{x1D705}_{lower}^{\infty }\rightarrow 0$ as $\unicode[STIX]{x1D6E5}/H\rightarrow \infty$ , contrary to the above result for a reversed triangle of $\unicode[STIX]{x1D705}_{lower}^{\infty }=0.549$ . To account for this expected deformation, the deflection function in the numerator of (4.4) is approximated by that of a trapezoidal plate with a point load at its leading edge, obtained using Euler–Bernoulli beam theory (Landau & Lifshitz Reference Landau and Lifshitz1970). The pressure is evaluated using the expected linearly varying mode shape away from the clamp (rigid body rotation pivoted at the clamp); this mode shape together with the evaluated pressure is used to calculate the denominator of (4.4). This analytical approach captures this asymptotic limit, and gives the following expression:

(4.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{lower}^{\infty }={\displaystyle \frac{24\left({\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}-1\right)^{3}}{\unicode[STIX]{x03C0}\left(4+5{\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}\right)\left[2\left({\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}\right)^{2}\log \left({\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}\right)+{\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}\left(4-3{\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}\right)-1\right]}},\quad {\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}\gg 1, & & \displaystyle\end{eqnarray}$$

which is also plotted in figure 6(a). The comparison in figure 6(a) shows that (4.8) provides a natural continuation to very large $\unicode[STIX]{x1D6E5}/H$ of the expression in (4.7), which is valid only up to moderately large $\unicode[STIX]{x1D6E5}/H$ . In the asymptotic limit $\unicode[STIX]{x1D6E5}/H\rightarrow \infty$ , equation (4.8) gives

(4.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{lower}^{\infty }\sim {\displaystyle \frac{24}{5\unicode[STIX]{x03C0}\left[2\log \left({\displaystyle \frac{\unicode[STIX]{x1D6E5}}{H}}\right)-3\right]}}, & & \displaystyle\end{eqnarray}$$

showing that an infinitesimal flow speed will lead to divergence in this limit, as expected.

4.3 Arbitrary planforms of finite aspect ratio

The above calculations do not account for the three-dimensional geometry of an inverted flag. To include the effect of finite generalised aspect ratio AR, we utilise a similar approach to that reported by Sader et al. (Reference Sader, Cossé, Kim, Fan and Gharib2016a ). For a flat sheet of arbitrary generalised aspect ratio $AR$ , it is well known that Prandtl’s lifting line theory (Jones Reference Jones1990; Anderson Reference Anderson2006) predicts a reduction in lift slope by factor of approximately $1+2/AR$ , ignoring the usual (small) Glauert parameter. This reduction in lift is due to the generation of side edge vortices that are always present on wings of finite aspect ratio. Therefore, the dimensionless (squared) flow speed $\unicode[STIX]{x1D705}_{lower}$ at which divergence occurs for an inverted flag of arbitrary shape and finite aspect ratio, relative to its infinite $AR$ value, $\unicode[STIX]{x1D705}_{lower}^{\infty }$ , is enhanced by this same factor, giving the result

(4.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{lower}\approx \unicode[STIX]{x1D705}_{lower}^{\infty }\left(1+{\displaystyle \frac{2}{AR}}\right), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D705}_{lower}^{\infty }$ for a trapezoidal morphology is specified by (4.7) and (4.8). Note that for $AR\sim O(1)$ , which is typical in the measurements reported in § 3, the value of $\unicode[STIX]{x1D705}_{lower}$ is approximately three times larger than $\unicode[STIX]{x1D705}_{lower}^{\infty }$ . This shows that side edge vortices play an important role in controlling the stability of inverted flags of finite aspect ratio.

To validate (4.10), a comparison of the predictions of (4.10) to those obtained using a VLM (Tornado 2015; Sader et al. Reference Sader, Cossé, Kim, Fan and Gharib2016a ) is given in figure 6(b). This is performed for sheets with $H/L=1.25$ , which corresponds to those studied experimentally in § 3. The generalised aspect ratio $AR$ (defined in (1.4)) for a trapezoidal sheet is

(4.11) $$\begin{eqnarray}\displaystyle AR\equiv 2{\displaystyle \frac{\text{max}\left(\unicode[STIX]{x1D6E5},H\right)^{2}}{\left(H+\unicode[STIX]{x1D6E5}\right)L}}. & & \displaystyle\end{eqnarray}$$

Vortex lattice method simulations involve calculating the lifting slope of a rigid trapezoidal sheet as a function of its shape parameter, $\unicode[STIX]{x1D6E5}/H$ ; they provide a more accurate and rigorous approach as compared to (4.10), if needed. The data in figure 6(b) show that (4.10) provides a good approximation. We note that lifting line theory – which is used to derive (4.10) – makes no distinction between swept and symmetric wings, and is normally applied to symmetric wings only (Jones Reference Jones1990; Anderson Reference Anderson2006). Even so, equation (4.10) captures the VLM simulations well. Note that the curves in figure 6(b) exhibit cusps at $\unicode[STIX]{x1D6E5}/H=1$ . This is consistent with experimental observations; see figures 4(a) and 7. This feature arises from the definition of the generalised aspect ratio for a trapezoid, equation (4.11), and our choice of fixing $H$ and $L$ , and varying $\unicode[STIX]{x1D6E5}/H$ .

4.4 Comparison of theory and measurement

Figure 7.

Comparison of dimensionless flow speed at divergence, $\unicode[STIX]{x1D705}_{lower}$ , obtained experimentally for two mass ratios: $\unicode[STIX]{x1D707}=0.40$ (▵) and $\unicode[STIX]{x1D707}=0.61$ (○) (see § 3), using the theory for infinite aspect ratio in (4.7) (dashed line) and with the finite aspect ratio correction in (4.10) (solid line). Error bars quantify the uncertainty in measuring $\unicode[STIX]{x1D705}_{lower}$ for a single sheet, and define the 25 % and 75 % points discussed in § 3.

Figure 7 provides a comparison of the measurement data of § 3 for the divergence instability (at which large-amplitude flapping begins) to the theory of §§ 4.1–4.3 in (4.7), (4.10) and (4.11). Data for two different flag thicknesses, and hence mass ratios $\unicode[STIX]{x1D707}=0.40$ and $0.61$ , are reported. This allows assessment of the expected independence of the divergence instability on $\unicode[STIX]{x1D707}$ .

Our theory for $\unicode[STIX]{x1D705}_{lower}$ exhibits the same qualitative behaviour as measurements and increases in a nonlinear fashion for decreasing shape parameter, $\unicode[STIX]{x1D6E5}/H$ . The 2-D strip model of (4.7) for infinite $AR$ strongly underestimates the measured data, but when corrected using (4.10) for finite $AR$ yields good agreement. Notably, this finite $AR$ theory slightly overestimates the measurements, which may be due to two independent effects. First, the strip theory of § 4.1 uses an approximate mode shape and thus strictly provides an upper bound on $\unicode[STIX]{x1D705}_{lower}^{\infty }$ , as discussed. This effect is expected to be small given the good agreement with the exact result for a rectangular planform. Second, the Prandtl lifting line correction in (4.10) implicitly assumes the vortex lines are parallel to the sheet. This is also an approximation and ignores the nonlinear lift generated by the true non-parallel nature of the vortex lines which increases lift (Bollay Reference Bollay1939; Sader et al. Reference Sader, Huertas-Cerdeira and Gharib2016b ). The omission of this (nonlinear lift) factor leads to (4.10) overestimating the true flow speed at which divergence occurs, as was demonstrated for an inverted flag of rectangular planform by Sader et al. (Reference Sader, Huertas-Cerdeira and Gharib2016b ). The present measurements for a trapezoidal planform are entirely consistent with Sader et al. (Reference Sader, Huertas-Cerdeira and Gharib2016b ), showing that the theoretical model for $\unicode[STIX]{x1D705}_{lower}$ in §§ 4.1–4.3 robustly predicts the divergence instability initiating large-amplitude flapping as a function of inverted flag morphology.

5.1 General observations

A sample of measurements on Flag B is given in figure 8, which shows stroboscopic images and time series of this flag’s motion as a function of the impinging flow speed, $U$ . All major behavioural regimes observed previously for inverted flags without a petiole are still found. Namely as the flow speed increases, the flag transitions from a straight (undeflected) mode to a flapping mode (periodic and chaotic), and finally, a deflected mode.

All flags exhibit deformation primarily in the petiole, i.e. the elastic sheet moves predominantly as a rigid (and straight) body. This is in contrast to flags without a petiole, where deformation occurs throughout the entire elastic sheet, cf. figures 1(c) and 8. These observations establish that the petiole can provide the primary source of elasticity to the flag when present, behaving as an elastic hinge. Reducing the petiole length, $L_{p}$ , or increasing the petiole rigidity beyond a critical value results in mechanical failure of the elastic sheet, where large stresses at its junction point with the petiole eventually tear the polycarbonate sheet over a few flapping cycles. This observation suggests that damage from flapping in an inverted orientation could impose structural constraints in real leaves. Indeed, it may be a reason why the petiole often extends into the leaf’s lamina and gradually tapers off (rather than abruptly ending), while veins into the lamina provide further structural support.

Figure 9(a) gives measurements of (i) the (peak-to-peak) amplitude, $A$ , at free end of the elastic sheet, and (ii) the (peak-to-peak) amplitude, $A_{p}$ , at the end of the petiole, for each flag as a function of the impinging flow speed, $U$ . The dynamics of the amplitudes, $A$ and $A_{p}$ , aligns with respect to the flow speed, $U$ . This is consistent with the flag deforming in a monotonically increasing fashion from its clamp to its free end, as seen in figure 8. Flag C, with the longest and most rigid petiole relative to its elastic sheet, requires the highest critical flow speed to initiate and cease flapping. This flag also displays the highest natural resonant frequency and flapping frequency. Interestingly, large-amplitude flapping in Flag E (that does not contain a petiole) is initiated and stops at comparable flow speeds to those of Flag C. At first sight, this appears to coincide with the flapping and natural resonant frequencies of these two flags being similar; see table 1. However, Flag A also contains similar frequencies but displays very different behaviour; see figure 9. This demonstrates that the inverted flag’s stability is not determined solely by the presence of a petiole nor by its flapping and natural resonant frequencies, with a more detailed analysis being required.

Figure 9.

Dimensionless (peak-to-peak) flapping amplitude of the elastic sheet’s free end (a,b) and at the petiole–sheet junction point (c,d) for Flag A (●), Flag B (○), Flag C (▴), Flag D (▵) and Flag E (dashed line). (a,c) Raw data plotted as a function of the dimensional flow velocity, $U$ . (b,d) Same data as in (a,c) plotted versus the rescaled dimensionless flow speed, $\hat{\unicode[STIX]{x1D705}}$ , defined in (5.5). This rescaling is theoretically predicted to collapse data for the initiation of large-amplitude flapping onto $\hat{\unicode[STIX]{x1D705}}=1$ , which is the divergence flow speed in (5.3); shown as vertical dash-dotted line. Experimentally measured values for the dimensionless flow speeds: $\unicode[STIX]{x1D705}_{lower}=0.038\pm 0.007$ , $0.050\pm 0.012$ , $0.106\pm 0.015$ and $0.124\pm 0.037$ for Flags A, B, C and D, respectively. Error bars quantify the uncertainty in measuring $\unicode[STIX]{x1D705}_{lower}$ , and define the 25 % and 75 % points discussed in § 3.

5.2 Mechanism for the initiation of flapping

The above observations that the petiole (i) is unaffected by the impinging flow, and (ii) provides the primary source of elasticity to the flag, suggests a decoupling of the hydrodynamic and elastic properties of these flags. This in turn enables the development of a simple theoretical model for the divergence instability that makes use of the lift generated by a rigid flat sheet at small angle of attack. It remains to be seen if such a divergence instability initiates large-amplitude flapping, as found for flags without a petiole. The theory developed in this section enables this issue to be explored.

The dimensional pressure jump across a flat sheet of length, $(L-L_{p})$ , at angle of attack, $\unicode[STIX]{x1D6FC}$ , to an impinging flow of speed, $U$ , follows from (4.1) and is given by

(5.1) $$\begin{eqnarray}\displaystyle p=2\unicode[STIX]{x1D70C}U^{2}\unicode[STIX]{x1D6FC}\sqrt{{\displaystyle \frac{L-L_{p}}{x}}-1}, & & \displaystyle\end{eqnarray}$$

where $x$ is the dimensional coordinate in the flow direction, whose origin is at the leading edge of the rigid sheet; see figure 3(a). This rigid sheet is attached to the petiole, a cantilevered elastic beam of length $L_{p}$ , whose deflection function, $w(x)$ , obeys the Euler–Bernoulli equation (Landau & Lifshitz Reference Landau and Lifshitz1970). It then follows that the force, $F$ , and moment, $M$ , exerted by the flat plate on the end of the petiole are

(5.2a,b ) $$\begin{eqnarray}\displaystyle F=\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}U^{2}H(L-L_{p})w^{\prime }(L_{p}),\quad M=-{\displaystyle \frac{3\unicode[STIX]{x03C0}}{4}}\unicode[STIX]{x1D70C}U^{2}H(L-L_{p})^{2}w^{\prime }(L_{p}), & & \displaystyle\end{eqnarray}$$

where the rigid sheet’s angle of attack, $\unicode[STIX]{x1D6FC}\equiv -w^{\prime }(L_{p})$ . The small tab-like overlap hinge connecting the sheet and petiole has its own bending stiffness, which is ignored in the present model, i.e. this hinge is assumed to be infinitely stiff.

The Euler–Bernoulli equation for this end-loaded beam is $w^{\prime \prime \prime \prime }(x)=0$ , with boundary conditions, $w(L)=w^{\prime }(L)=0$ , $w^{\prime \prime }(L-L_{p})=-M/EI$ and $w^{\prime \prime \prime }(L-L_{p})=F/EI$ , where $EI=Eh_{p}^{3}H_{p}/12$ . Solving this eigenvalue problem gives the required dimensionless flow speed for which divergence occurs,

(5.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{lower}^{pet}\equiv {\displaystyle \frac{\unicode[STIX]{x1D70C}U_{lower}^{2}L_{p}^{2}\left(L-L_{p}\right)}{D^{\prime }}}\approx {\displaystyle \frac{4}{\unicode[STIX]{x03C0}}}\left(1+{\displaystyle \frac{2}{AR}}\right){\displaystyle \frac{L_{p}}{3L-L_{p}}}{\displaystyle \frac{H_{p}}{H}}, & & \displaystyle\end{eqnarray}$$

where $D^{\prime }\equiv Eh_{p}^{3}/12$ is the flexural rigidity of the petiole, $U_{lower}$ is the critical flow speed for divergence and the Prandtl’s lifting line correction, (4.10), for the sheet’s finite generalised aspect ratio, $AR$ , is included, with

(5.4) $$\begin{eqnarray}\displaystyle AR={\displaystyle \frac{H}{L-L_{p}}}. & & \displaystyle\end{eqnarray}$$

We emphasise that (5.3) assumes that the flag’s elasticity is isolated to the petiole; see above.

Figure 9(b) presents the same data as in figure 9(a) but now plotted as a function of the rescaled dimensionless flow speed,

(5.5) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D705}}\equiv {\displaystyle \frac{\unicode[STIX]{x1D705}^{\prime }}{\unicode[STIX]{x1D705}_{lower}^{pet}}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D705}^{\prime }\equiv \unicode[STIX]{x1D70C}U^{2}L_{p}^{2}(L-L_{p})/D^{\prime }$ . Equation (5.5) implicitly contains the divergence instability’s geometric dependence in (5.3) for an inverted flag with a petiole; divergence is theoretically predicted to occur at $\hat{\unicode[STIX]{x1D705}}\approx 1$ . Data for Flag E are omitted from figure 9(b) because this case does not have a petiole.

Figure 10.

Stroboscopic images of Flags A–D in their respective flapping regimes at various flow speeds, which are selected so that the flags exhibit similar flapping amplitudes. Flag A: $U=4.77$  m s $^{-1}$ . Flag B: $U=4.11$  m s $^{-1}$ . Flag C: $U=6.09$  m s $^{-1}$ . Flag D: $U=4.77$  m s $^{-1}$ . Flag A exhibits a notable kink where the petiole connects the elastic sheet, which is smaller than that for Flag B (as indicated by dashed lines). Flags C and D display significant twisting in their elastic sheets, which is manifest by an apparent discontinuity at the connection between the sheet and petiole – the images are taken in the gravity direction.

This rescaled plot shows that (5.3) accurately captures the divergence instability for Flag A – with data collapsing onto $\hat{\unicode[STIX]{x1D705}}\approx 1$ in the flow speed region where large-amplitude flapping is initiated (at low flow speed). This rescaling also works well for Flag B, although divergence (and the initiation of flapping) occurs at a slightly higher value of $\hat{\unicode[STIX]{x1D705}}$ . The reason for this difference is evident in stroboscopic images of these flags in the flapping regime; see figure 10. While the theory derived in this section assumes the deflection angle of the petiole and elastic sheet are identical at their connection position, this is not satisfied exactly in measurements: Flags A and B displaying different behaviours at this connection point. The thicker (and hence more rigid) the petiole the stronger the discontinuity, which leads to a larger violation of the theory’s assumptions for Flag B relative to Flag A; the hinge connecting the sheet and petiole can deform to a greater degree, violating the model’s assumption of an infinitely stiff hinge. Even so, this is found to exert a small effect with the derived approximate theory performing well and accurately capturing the divergence instability for both Flags A and B with no fitting parameters. We emphasise that the theory contains a number of approximations – in addition to the rigid hinge assumption – and therefore precise agreement between measurement and theory is not expected.

In contrast, the divergence instability and onset of flapping is underestimated by the developed theory for Flags C and D, with divergence occurring at $\hat{\unicode[STIX]{x1D705}}\approx 2$ ( ${>}$ 1). The stroboscopic images in figure 10 also provide some insight into this difference, with strong twisting behaviour in Flags C and D. This is manifest in an apparent discontinuity in the displacement between the petiole and the sheet, which is greatly reduced in Flags A and B. Such behaviour, due to the coupling effects of gravity and twisting of the petiole (the sheet deforms in a rigid fashion with the inverted flag displaying a waving motion as it flaps), is not included in (5.3) and appears to be causing the divergence instability to occur at twice the theoretical value. Even so, increasing the petiole thickness enhances the value of $\hat{\unicode[STIX]{x1D705}}$ at which divergence is observed, as for Flags A and B. Similar twisting behaviour has been previously observed for slender inverted flags (Sader et al. Reference Sader, Huertas-Cerdeira and Gharib2016b ). Reducing the petiole length enhances its torsional stiffness and decreases this behaviour; see figure 10. Finally, we note that the observed discrepancies between theory and measurement for Flags C and D may also be related to differences in flow processes, e.g. vortex formation. Flow visualisation remains an avenue for future work.

For Flags A–D, the undeflected mode of the inverted flag abruptly gives way to large-amplitude flapping. This shows that a divergence instability triggers this flapping instability, as observed for inverted flags without a petiole. Finally, we note that the rescaling in (5.5) does not collapse data at higher flow speed where flapping ceases and the deflected mode appears. Thus, the empirical relation of $\unicode[STIX]{x1D705}_{upper}/\unicode[STIX]{x1D705}_{lower}\approx 4$ observed for trapezoidal inverted flags with no petiole (§ 3) does not hold for rectangular flags with petioles. Adding a petiole to an inverted flag can therefore enable adjustment of the relative flow speed range over which large-amplitude flapping occurs.