Flapping States of a Flag in an Inviscid Fluid: Bistability and the Transition to Chaos

We investigate the “flapping flag” instability through a model for an inextensible flexible sheet in an inviscid 2D flow with a free vortex sheet. We solve the fully-nonlinear dynamics numerically and find a transition from a power spectrum dominated by discrete frequencies to an apparently continuous spectrum of frequencies. We compute the linear stability domain which agrees with previous approximate models in scaling but differs by large multiplicative factors. We also find hysteresis, in agreement with previous experiments.

Figure 1

Flag and wake dynamics for $R_1=0.3$ and $R_2=0.01445$. The time-averaged flow velocity field of the flag from $t=0$ to $t=120$ is shown (red arrows: average over 60 flapping periods), with the background velocity removed. The flag position ($C_b$, in black, with normal $\widehat{\mathbf{n}}$ and tangent $\widehat{\mathbf{s}}$ labeled) and trailing vortical wake (the free-sheet $C_f$, blue-dashed line) is shown at $t=120$.

Figure 2

Snapshots of the flag for fixed mass ($R_1=0.3$) and decreasing rigidities $R_2$. (a) The observed flapping mode at $R_2=0.01445$, about a factor of 2 below the critical $R_2=0.0262$; (b) a higher energy flapping mode for $R_2=0.0138$; (c) a chaotic flapping mode at $R_2=0.0025$. (d) Comparison of experimental flag snapshots in 18, Fig. 12b, with model shapes. In both cases $R_1=0.37$ (see text for $R_2$).

Figure 3

Temporal power spectra of flag bending energy vs. frequency, for $R_1=0.3$ and $R_2$ equal to 0.014 45 (a), 0.014 36 (b), 0.0138 (c), and 0.0025 (d).

Figure 4

Computed stability boundary in the $R_1{R}_2$ plane. The upper solid boundary gives the smallest $R_2$ above which a small leading-edge forcing ($y(t)={10}^{-5}(2t{)}^2{e}^{-(2t{)}^2}$) does not lead to flapping. The lower solid boundary is the largest $R_2$ below which such forcing leads to exponential growth of elastic energy in time until the flag saturates with $O(1)$ flapping, as shown in Figs. 1, 2. The solid line gives the scaling $R_1\sim R_2$ for comparison at small flag masses. The black crosses mark the cases shown in Fig. 2 [upper cross is (a) and (b), and lower is (c)]. The dashed line shows the stability boundary from [8], and the dash-dotted line the corresponding boundary plotted in Fig. 3 of 7 (showing only the portion $R_2\approx \mathrm{\text{const}}$, but having the same asymptotic scalings as the model here and in 8).

Figure 5

Bistability and hysteresis in the onset of flapping with changes of flag rigidity $R_2$. (a) For $R_1=0.03$ and $R_2=2.74\times {10}^{-3}$, four different values of transverse perturbations $A_0(2t{)}^2{e}^{-(2t{)}^2}$ are applied to the flag leading edge, for $A_0$ in the range 0.06 to 0.4, and bending energy is plotted in time. The hysteresis loop for $R_1=0.03$ is shown in (b), with the flat and steady flapping states in (a) circled. The time-averaged bending energy in the asymptotic steady state is plotted. (c) The hysteresis loop for mass $R_1=0.3$, an order of magnitude larger than in (b), showing a range of bistability which is also an order of magnitude larger.