Eigenmode analysis of membrane stability in inviscid flow

We study the instability of a thin membrane (of zero bending rigidity) to out-of-plane deflections, when the membrane is immersed in an inviscid fluid flow and sheds a trailing vortex-sheet wake. We solve the nonlinear eigenvalue problem iteratively with large ensembles of initial guesses for three canonical boundary conditions—both ends fixed, one end fixed and one free, and both free. Over several orders of magnitude of membrane mass density, we find instability by divergence or flutter (particularly at large mass density or with one or both ends free). The most unstable eigenmodes generally become wavier at smaller mass density and smaller tension but with regions of nonmonotonic behavior. We find good quantitative agreement with unsteady time-stepping simulations at small amplitudes, but only qualitative similarities with the eventual steady-state large-amplitude motions.

Figure 1

Schematic diagram of flexible membranes (solid curved black lines) at an instant in time. Here $2L$ is the chord length (the distance between the endpoints) and $U$ is the oncoming flow velocity. For the nonlinear, large-amplitude model [(a), (c), (e)], $y(x,t)$ is the membrane deflection and the dashed line is the free vortex wake. The right columns show the corresponding linearized, small-amplitude eigenvalue problems, where the motions are represented by the real and imaginary parts of the eigenmodes $Y\left(x\right)$, shown as green and blue lines, respectively, and flat vortex wakes of fixed length $L_w$ shown as dashed lines at $y=0$ [(b), (d), (f)]. The boundary conditions shown are: fixed-fixed membranes [(a), (b)], fixed-free membranes [(c), (d)], and free-free membranes [(e), (f)].

Figure 2

Fixed-fixed eigenvalues and eigenmodes with $R_1={10}^{-1}$ and $T_0={10}^{-0.27}$. Computed ${\sigma }_{\mathrm{R}}$ [(a), values in color bars at right] and computed ${\sigma }_{\mathrm{I}}$ [(b), values in color bars at right], both plotted over the initial guess complex plane (c). The distinct eigenvalues generated by the numerical method replotted as red dots in the $({\sigma }_{\mathrm{R}},{\sigma }_{\mathrm{I}})$ plane. (d) The corresponding eigenmodes [$\mathrm{Re}\left(Y\right(x\left)\right)$ in green, $\mathrm{Im}\left(Y\right(x\left)\right)$ in blue] from the only unstable one (with negative ${\sigma }_{\mathrm{I}}$) on the left to the most stable one (largest ${\sigma }_{\mathrm{I}}$) on the right. The vertical black line separates the unstable mode (on its left) and stable modes (on its right).

Figure 3

Fixed-fixed eigenvalues and eigenmodes with $R_1={10}^3$ and $T_0={10}^{1.5}$ and other quantities as described in Fig. 2.

Figure 4

The region in $R_1$–$T_0$ space in which the fixed-fixed membrane is unstable. The red line and red dots indicate the position of the stability boundary computed by linear interpolation between ${\sigma }_{\mathrm{I}}$ of the smallest $T_0$ that gives a stable membrane and the ${\sigma }_{\mathrm{I}}$ of the largest $T_0$ that gives an unstable membrane (shown in the error bars). The colors of the dots below the stability boundary label: (a) The imaginary parts of the eigenvalues (${\sigma }_{\mathrm{I}}$) corresponding to the most unstable modes. They represent the temporal growth rate. (b) The real parts of the eigenvalues (${\sigma }_{\mathrm{R}}$) for the most unstable modes, representing the angular frequencies. The gray dots correspond to modes that lose stability by divergence and have ${\sigma }_{\mathrm{R}}\le {10}^{-9}$. The orange line that spans ${log}_{10}{R}_1\in [-2,2]$ represents the stability boundary computed numerically in Ref. [1].

Figure 5

The shapes $Y\left(x\right)$ of the most unstable eigenmodes as a function of $R_1$ and $T_0$ in the fixed-fixed case. The real and imaginary parts of $Y\left(x\right)$ are shown in green and blue, respectively. Each shape is scaled both vertically and horizontally to fit within the plot. The shapes are superposed on the same stability boundary (red line) as in Fig. 4. Modes exhibiting a divergence instability have a gray rectangle outline.

Figure 6

For two values of membrane mass ($R_1$), ${10}^3$ (left column) and ${10}^1$ (right column), the imaginary [(a), (c)] and real parts [(b), (d)] of the eigenvalues versus pretension ($T_0$) for fixed-fixed membranes. The coloring represents the RMS of the membrane's slope, $Y_{\mathrm{RMS}}^{\prime }$ (32) for each ($R_1,T_0$) pair. The horizontal black lines in the top panels located at (a) ${\sigma }_{\mathrm{I}}=\pm {10}^{-6}$ and (c) ${\sigma }_{\mathrm{I}}=\pm 3\times {10}^{-4}$ distinguish stable modes (above) and unstable modes (below). To the left of and within (a) and (c), we show typical modes for branches with $Y_{\mathrm{RMS}}^{\prime }<4\pi$.

Figure 7

The region in $R_1$–$T_0$ space in which the fixed-free membrane is unstable. The red line and red dots indicate the position of the stability boundary computed using linear interpolation between ${\sigma }_{\mathrm{I}}$ of the smallest $T_0$ that gives only stable eigenmodes and the ${\sigma }_{\mathrm{I}}$ of the largest $T_0$ that gives an unstable eigenmode (shown in the error bars). The color of the dots below the stability boundary labels: (a) The imaginary part of the eigenvalue (${\sigma }_{\mathrm{I}}$) corresponding to the most unstable modes. It represents the temporal growth rate. (b) The real part of the eigenvalues (${\sigma }_{\mathrm{R}}$) for the most unstable mode, representing the angular frequency. The orange line that spans $R_1\in [{10}^{-3},{10}^2]$ represents the stability boundary computed numerically in Ref. [1].

Figure 8

Fixed-free eigenvalues and eigenmodes with $R_1={10}^3$ and $T_0={10}^{0.8}$. Computed ${\sigma }_{\mathrm{R}}$ [panel (a), values in color bars at right] and computed ${\sigma }_{\mathrm{I}}$ [(b), values in color bars at right], both in the initial guess complex plane. (c) The computed eigenvalues replotted as red dots in the $({\sigma }_{\mathrm{R}},{\sigma }_{\mathrm{I}})$ plane. The inset in (c) shows the ten computed eigenvalues (red $\circ$) that correspond to the eigenmodes shown in (d). The analytical form of the eigenvalues is $\sigma =((n-1/2)\pi /2)\sqrt{T_0/R_1}=((n-1/2)\pi /2)\sqrt{{10}^{0.8}/{10}^3}$ for $n=1,\cdots ,10$ (black pluses).

Figure 9

The shapes $Y\left(x\right)$ of the most unstable eigenmode as a function of $R_1$ and $T_0$ in the fixed-free case. The real part of $Y\left(x\right)$ is shown in green and the imaginary part of $Y\left(x\right)$ is shown in blue. Each shape is scaled, both vertically and horizontally, to fit within the plot. The shapes are superposed on the same stability boundary (red line) as in Fig. 7.

Figure 10

For two values of membrane mass ($R_1$), ${10}^3$ (left column) and ${10}^1$ (right column), the imaginary [(a), (c)] and real parts [(b), (d)] of the eigenvalues versus pretension ($T_0$) for fixed-free membranes. The coloring represents the RMS of the membrane's slope, $Y_{\mathrm{RMS}}^{\prime }$, for each ($R_1,T_0$) pair given by Eq. (32). The horizontal black line in the top panels located at (a) ${\sigma }_{\mathrm{I}}=\pm {10}^{-6}$, (c) ${\sigma }_{\mathrm{I}}=\pm {10}^{-4}$ distinguishes stable modes (above) and unstable modes (below). We also show typical modes that correspond to each branch with $Y_{\mathrm{RMS}}^{\prime }<9\pi /2$.

Figure 11

The region in $R_1$–$T_0$ space in which the free-free membrane is unstable. The red line and red dots indicate the position of the stability boundary computed using linear interpolation between ${\sigma }_{\mathrm{I}}$ of the smallest $T_0$ that gives a stable membrane and the ${\sigma }_{\mathrm{I}}$ of the largest $T_0$ that gives an unstable membrane (shown in the error bars). The color of the dots below the stability boundary labels: (a) The imaginary part of the eigenvalue (${\sigma }_{\mathrm{I}}$) corresponding to the most unstable modes. It represents the temporal growth rate. (b) The real part of the eigenvalues (${\sigma }_{\mathrm{R}}$) for the most unstable mode, representing the angular frequency. The orange line that spans $R_1\in [{10}^{-3},{10}^2]$ represents the stability boundary computed numerically in Ref. [1].

Figure 12

Free-free eigenvalues and eigenmodes with $R_1={10}^3$ and $T_0={10}^{1.1}$. Computed ${\sigma }_{\mathrm{R}}$ [(a), values in color bars at right] and computed ${\sigma }_{\mathrm{I}}$ [(b), values in color bars at right], both plotted in the initial guess complex plane. (c) The distinct eigenvalues generated by the numerical method plotted as red dots in the $({\sigma }_{\mathrm{R}},{\sigma }_{\mathrm{I}})$ plane. The analytical form of the eigenvalues is $\sigma =((n-1)\pi /2)\sqrt{T_0/R_1}$ for $n=1,\cdots ,46$ (black pluses). (d) The 11 lowest wave-number eigenmodes [$\mathrm{Re}\left(Y\right(x\left)\right)$ in green, $\mathrm{Im}\left(Y\right(x\left)\right)$ in blue], from the most unstable (most negative ${\sigma }_{\mathrm{I}}$) on the left to the most stable (largest positive ${\sigma }_{\mathrm{I}}$) on the right. The vertical black line separates unstable modes (on its left) and stable modes (on its right).

Figure 13

The shapes $Y\left(x\right)$ of the most unstable eigenmode as a function of $R_1$ and $T_0$ in the free-free case. The real part of $Y\left(x\right)$ is shown in green and the imaginary part of $Y\left(x\right)$ is shown in blue. Each shape is scaled, both vertically and horizontally, to fit within the plot. The shapes are superposed on the same stability boundary (red line) as in Fig. 11.

Figure 14

For two values of membrane mass ($R_1$), ${10}^3$ (left column) and ${10}^1$ (right column), the imaginary [(a), (c)] and real parts [(b), (d)] of the eigenvalues versus pretension ($T_0$) for free-free membranes. Numerical results are shown as points with color coded according to the value of the RMS of the membrane's slope for each ($R_1,T_0$) pair given by Eq. (32), as given in the color bar. The horizontal black line in the top panels located at (a) ${\sigma }_{\mathrm{I}}=\pm {10}^{-6}$, (c) ${\sigma }_{\mathrm{I}}=\pm {10}^{-4}$ distinguishes stable modes (above) and unstable modes (below). We also show typical modes that correspond to each branch with $Y_{\mathrm{RMS}}^{\prime }<9\pi /2$.

Figure 15

Fixed-fixed membranes at $R_1={10}^{-1}$ and (a) $T_0={10}^{-0.1}$, (b) $T_0={10}^0$, (c) $T_0={10}^{0.1}$, and (d) $T_0={10}^{0.2}$. These membranes lose stability by divergence. We compare the most unstable modes obtained from the eigenvalue analysis (dashed green lines) to the membrane shapes of the time-stepping simulations in the small-amplitude (growth) regime—in each panel, 15 equally spaced snapshots are shown in the growth regime, gray and then black at the last time. The arbitrary amplitudes of the green lines are set to match those of the black lines. The light blue curves indicate shapes in the large-amplitude steady state regime.

Figure 16

Fixed-free membranes at $R_1={10}^{-0.5}$ and $R_3={10}^{1.5}$, with $T_0={10}^{-0.8}$ for (a), (c), (e) and $T_0={10}^{-0.7}$ for (b), (d), (f). In (a) and (b), the solid red lines are $\mathrm{Re}\left(y_{\mathrm{nonlin}}\left(\alpha \right)\right)$ estimated from the time-stepping simulation, which are close to $\mathrm{Re}\left(Y\right(x\left)\right)$ from the eigenvalue problem (dotted black lines). The solid green lines are $\mathrm{Im}\left(y_{\mathrm{nonlin}}\left(\alpha \right)\right)$, close to $\mathrm{Im}\left(Y\right(x\left)\right)$ from the eigenvalue problem (dotted blue lines). The gray lines are a subset of snapshots in the linear growth regime. In (c) and (d), we show snapshots during the small-amplitude (growth) regime, but with the exponential growth removed. (e), (f) show snapshots during the steady-state large-amplitude motions. We show 20 equally spaced snapshots of membranes over a period, ranging from light blue at earlier times to dark blue at the last time.

Figure 17

Same quantities as described in Fig. 16 but with $R_1={10}^1$ and $R_3={10}^{1.5}$, and $T_0={10}^{-0.1}$ [for (a), (c), and (e)]; $T_0={10}^0$ [for (b), (d) and (f)].

Figure 18

Comparison between the (a) real and (b) imaginary parts of the eigenmodes with fixed-free boundary conditions, using grids with $m=80$ and $m=120$. Each shape is scaled in both vertical and horizontal directions to fit within the plot. The red dots indicate the position of the stability boundary (same as in Fig. 9).

Figure 19

At each $(R_1,T_0)$ in the instability region (below red line), the relative error Eq. (A2) in the eigenvalues when using $m=80$ and $m=120$ Chebyshev points on the fixed-free membrane is plotted as a colored dot.

Figure 20

Spectrum of eigenvalues for $m=80$ (green diamonds), 120 (red circles), and 240 (blue crosses) for a fixed-free membrane at (a) $(R_1,T_0)=({10}^{0.5},{10}^{-0.25})$, (b) $(R_1,T_0)=({10}^3,{10}^{0.8})$, and a free-free membrane at (c) $(R_1,T_0)=({10}^{0.5},{10}^{-0.5})$.

Figure 21

Example of the comparison method using data from fixed-free membranes with $(R_1,T_0)=({10}^{-0.5},{10}^{-0.7})$. (a) $ln\left(\right|y\left|\right)$ versus time for the 10th (blue), 30th (red), and 100th (yellow) grid points on the membrane. (b) A portion of the time series of $y{e}^{{\sigma }_{\mathrm{I}}t}$ at the 10th (blue), 30th (red), and 100th (yellow) grid points. This corresponds to part of the small-amplitude regime but with the growth removed. The black dashed lines represent the constructed $R\left(\alpha \right)cos({\sigma }_{\mathrm{R}}t+\varphi \left(\alpha \right))$ at the same grid points. (c) The reconstructed data $\mathrm{Re}\left([\mathrm{Re}\left(y_{\mathrm{nonlin}}\left(\alpha \right)\right)+i\mathrm{Im}\left(y_{\mathrm{nonlin}}\left(\alpha \right)\right)]e^{i\sigma t}\right)$ (black dashed lines) compared against the data $y(\alpha ,t)$ (cyan solid lines) at the times $t=20,100,160$. The initial perturbation here is $\zeta (\alpha ,0)=\eta sin\left(\pi \alpha \right)$ where $\eta$ is chosen as 0.0001. Note that the axes are not to scale.

Figure 22

Free-free membranes at $(R_1,T_0)=({10}^1,{10}^{0.1})$ for (a), (c), and (e) and at $(R_1,T_0)=({10}^{1.5},{10}^{0.2})$ for (b), (d), and (f), with $R_3={10}^{1.5}$ in both cases. These membranes lose stability by flutter and divergence. In (a) and (b), the solid red lines are $\mathrm{Re}\left(y_{\mathrm{ynonlin}}\left(\alpha \right)\right)$ estimated from the time-stepping simulation, which are close to $\mathrm{Re}\left(Y\right(x\left)\right)$ from the eigenvalue problem (dotted black lines). The solid green lines are $\mathrm{Im}\left(y_{\mathrm{ynonlin}}\left(\alpha \right)\right)$, close to $\mathrm{Im}\left(Y\right(x\left)\right)$ from the eigenvalue problem (dotted blue lines). The gray lines are a subset of snapshots in the linear growth regime. In (c) and (d), we show the snapshots during the small-amplitude (growth) regime but with the exponential growth removed. (e) and (f) show snapshots during the steady-state large-amplitude motions. Shades of gray (and blue) increase from light to dark as 20 membrane positions cycle through a period.