Dynamics of flags over wide ranges of mass and bending stiffness

There have been many studies of the instability of a flexible plate or flag to flapping motions, and of large-amplitude flapping. Here we use inviscid simulations and a linearized model to more generally study how key quantities—mode number (or wave number), frequency, and amplitude—depend on the two dimensionless parameters: flag mass and bending stiffness. In the limit of small flag mass, flags perform traveling wave motions that move at nearly the speed of the oncoming flow. The flag mode number scales as the $-1/4$ power of bending stiffness. The flapping frequency has the same scaling, with an additional slight increase with flag mass in the small-mass regime. The flapping amplitude scales approximately as flag mass to the 1/2 power. For large flag mass, the dominant mode number is low (0 or 1), the flapping frequency tends to zero, and the amplitude saturates in the neighborhood of its upper limit (the flag length). In a linearized model, the fastest growing modes have somewhat different power law scalings for wave number and frequency. We discuss how the numerical scalings are consistent with a weakly nonlinear model.

Figure 1

Example of flag snapshots (gray and green lines) and vortex sheet wake (blue line) for $R_1={10}^{-2.5}$ and $R_2={10}^{-5.05}$. The green and blue lines are the flag and wake positions at $t=9.5$. The gray lines are the flag positions at six equally spaced times from $t=5.375$ to 9.0 in increments of 0.725.

Figure 2

Sets of flag snapshots at various combinations of flag mass $\left(R_1\right)$ and bending stiffness $\left(R_2\right)$. Each set of snapshots is scaled vertically to enhance visibility. The true maximum and minimum vertical deflections (relative to the horizontal scales of the flags) are shown by the black scale bars to the left of each set of snapshots. Beneath each of column of flags is a column of dots of the same color, at the corresponding $(R_1,R_2)$ values.

Figure 3

Measures of flag dynamics across ($R_1, R_2$) space. (a) Mean mode number, defined by (11), the time average of the dominant Fourier component of the flag deflection. (b) Mean flapping frequency, defined as twice the reciprocal of the time-average duration between sign changes of the free end deflection. (c) Mean flapping amplitude, defined as the root-mean-square deflection of the flag's free end.

Figure 4

Power law behaviors of measures of flag dynamics. (a) Mean flag mode number vs $R_2$, for various $R_1$ (labeled by color; see legend at far right). (b) Mean flapping frequency vs $R_2$, for various $R_1$. (c) Flapping amplitude (root-mean-square vertical displacement at the free end) vs $R_2$, for various $R_1$. (d) Flapping amplitude vs $R_1$, for various $R_2$.

Figure 5

Flag deflections in the rest frame of the far-field flow. Plotted here are the topmost sets of flag snapshots from the left seven columns in Fig. 2, with ${10}^{-4.5}\le R_1\le {10}^{-1.5}$, with respect to $x-t$ on the horizontal axis (the rest frame of the fluid in the far field). Each set of snapshots is scaled vertically to make the deflections visible.

Figure 6

The two regimes of asymptotic scalings in the instability region for the spatially periodic, small-amplitude model. The blue dash-dotted line $R_1=R_2^3$ is approximately the location of the transition region between the two regimes. In each regime, the scalings of the fastest growing mode's wave number $k_{max}$, corresponding growth rate ${\sigma }_{I,k_{max}}$, and angular frequency ${\sigma }_{R,k_{max}}$ are listed. The light-blue and light-green lines show examples of stability boundaries that correspond to $k=2\pi$ (light-green dashed line) and $k=1$ (light-blue dashed line).

Figure 7

Comparison between the dominant Fourier modes (blue) and flag snapshots (gray, green, yellow, orange, and red) corresponding to the topmost sets of flag snapshots from the left seven columns in Fig. 2, with ${10}^{-4.5}\le R_1\le {10}^{-1.5}$. These are also the snapshots in Fig. 5. In the left two columns, only the trailing halves of the flags are shown to enhance visibility.

Figure 8

Plots of mean mode number ($K$), frequency ($f$), and amplitude ($A$) vs $R_2$ (on horizontal axis). Each column of three subpanels has the same $R_1$ value (labeled at the top or bottom, in color). In the subpanels for $K$ and $f$ with ${10}^{-4}\le R_1\le {10}^{0.5}$, black lines show the scaling $R_2^{-1/4}$. In the subpanels for $A$ with ${10}^{-2}\le R_1\le {10}^{-1}$, black lines show the scaling $R_2^{1/4}$.