On the mechanism of flutter of a flag

Abstract

This paper is concerned with the dynamics and stability of a flapping flag, with emphasis on the onset of flutter instability. The mathematical model is based on the one derived in a paper by Argentina and Mahadevan (Proc Nat Acad Sci 102:1829–1834, 2005). In that paper, it is reported that the effect of vortex shedding from the trailing edge of the flag, represented by the complex Theodorsen function C, has a stabilizing effect, in the sense that the critical flow speed (where flutter is initiated) is increased significantly when vortex shedding is included. The numerical eigenvalue analyses of the present paper display the opposite effect: the critical flow speed is decreased when the Theodorsen function (i.e., vortex shedding) is included. These predictions are verified by an analytical energy balance analysis, where it is proved that a small imaginary part of the Theodorsen function, $C = 1 - \mathrm {i} \, \epsilon$, $0 < \epsilon \ll 1$, has a destabilizing effect, i.e., the critical flow speed is smaller than by the so-called quasi-steady approximation $C = 1 - \mathrm {i} \, 0$. Furthermore, order-of-magnitude considerations show that Coriolis and centrifugal force terms in the equation of motion, previously discarded on the assumption that they are associated with very slow changes across the flag, have to be retained. Numerical results show that these terms have a significant effect on the stability of the flag; specifically, the said destabilizing effect of the vortex shedding is significantly reduced when these terms are retained. The mentioned energy balance analysis illuminates the nature of the flutter oscillations and the ‘competition’, at the flutter threshold, between the different types of fluid forces acting on the flag.

Appendices

Convergence

Table 1 shows the convergence of the critical flow speed $u_c$ and corresponding flutter frequency $\omega _c$ in terms of the number of finite elements $N_e$. The results in columns 1–4 are for the quasi-steady case with $C(\kappa ) \equiv 1 - \mathrm {i}\,0$, for the two different mass ratios $\rho = 0.2$ and $\rho = 25$, without inclusion of Coriolis and centrifugal force terms. It is seen that the convergence is more rapid for $\rho = 25$ than for $\rho = 0.2$. Columns 5–8 (marked with asterisks) are for the mass ratios $\rho = 0.2$ and $\rho = 2$, based on the ‘full’ equation of motion (7), with inclusion of the Coriolis and centrifugal force terms and with proper inclusion of the Theodorsen function $C(\kappa )$. By comparison with the first two columns, it is seen that the convergence speed is not significantly affected by inclusion of the Theodorsen function and the Coriolis and centrifugal force terms.

Table 1 Convergence of the critical flow speed $u_c$ and corresponding flutter frequency $\omega _c$ in terms of the number of finite elements $N_e$

Asymptotic values of the Theodorsen function

The asymptotic limits $\kappa \rightarrow 0$ and $\kappa \rightarrow \infty$ are investigated here for the Theodorsen function

$$\begin{aligned} C(\kappa ) = \frac{H_1^{(2)}(\kappa )}{H_1^{(2)}(\kappa ) + \mathrm {i} H_0^{(2)}(\kappa )} \end{aligned}$$

(40)

with the argument $\kappa = \omega /2 u_0$. For $\kappa \ll 1$, it is found that [1]

$$\begin{aligned} H_0^{(2)}(\kappa )= 1 - \frac{\kappa ^2}{2^2} + O\big (\kappa ^4\big ) - \mathrm {i} \left[ \frac{2}{\pi } \bigg \{ \ln \left( \frac{\kappa }{2} \right) + \gamma \bigg \} \bigg \{ 1 + \frac{\kappa ^2}{2^2} + O\big (\kappa ^4\big ) \bigg \} + \frac{2}{\pi } \bigg \{ \frac{\kappa ^2}{2^2} + O\big (\kappa ^4\big ) \bigg \} \right] , \end{aligned}$$

(41)

and

$$\begin{aligned} H_{1}^{(2)}(\kappa ) = \frac{\kappa }{2} \Bigg [ 1 - \frac{(\kappa /2)^2}{1\,!\,2\,!} + O\big (\kappa ^4\big ) \Bigg ] + \mathrm {i} \Bigg [ \frac{2}{\pi } \frac{1}{\kappa } + \frac{\kappa }{\pi } \bigg \{ 1 + \gamma - \ln \left( \frac{\kappa }{2} \right) \bigg \} + O\big (\kappa ^3 \ln \kappa \big ) \Bigg ], \end{aligned}$$

(42)

where $\gamma$ is Euler’s constant, $\gamma = 0.5771 \dots$. These results give that

$$\begin{aligned} C(\kappa ) = 1 - \frac{\pi }{2}\kappa + O\big (\kappa ^2 \ln \kappa \big ) - \mathrm {i} \bigg [ \kappa \big \{ \ln 2 - \gamma - \ln \kappa \big \} + O\big (\kappa ^2 \ln \kappa \big ) \bigg ] \end{aligned}$$

(43)

for $\kappa \ll 1$. We can go a step further and consider the following series expansion of $\ln \kappa$ [1], valid for $|\kappa - 1| \le 1$:

$$\begin{aligned} \ln \kappa = (\kappa - 1) - {\textstyle \frac{1}{2}}(\kappa - 1)^2 + \cdots . \end{aligned}$$

Then, (43) can be written as

$$\begin{aligned} C(\kappa ) = 1 - \frac{\pi }{2}\kappa + O\big (\kappa ^2 \ln \kappa \big ) - \mathrm {i} \Big [ \kappa \big \{ \ln 2 + 1 - \gamma - \kappa \big \} + O\big (\kappa ^3 \big ) \Big ]. \end{aligned}$$

(44)

For $\kappa \gg 1$, $H_0^{(2)}(\kappa )$ and $H_1^{(2)}(\kappa )$ can be expressed in unified form as [7]

$$\begin{aligned} H_{\nu }^{(2)}(\kappa ) \sim \left( \frac{2}{\pi \kappa }\right) ^{\frac{1}{2}} \mathrm {e}^{-\mathrm {i}(\kappa - \nu \pi /2 - \pi /4 )} \bigg [ 1 - \mathrm {i}\frac{\left( \nu ^2 - 1/4 \right) }{2\kappa } + O\big (\kappa ^{-2} \big ) \bigg ]. \end{aligned}$$

(45)

This result gives, with $\nu = 0$ and 1, that

$$\begin{aligned} C(\kappa ) \sim \frac{1}{2} \bigg \{1 + O\big (\kappa ^{-2} \big ) \bigg \} - \mathrm {i} \frac{1}{8\kappa } \bigg \{1 + O\big (\kappa ^{-2} \big ) \bigg \} \end{aligned}$$

(46)

for $\kappa \gg 1$.

Variation in wake wavenumber and in Theodorsen function with increasing flow speed

Figure 15 shows the wake wavenumber $\kappa = \omega _{*}/2 u$ as function of the flow speed u for $\rho = 0.2$, corresponding to the eigenvalue curves shown in Fig. 7 (without the Coriolis and centrifugal force terms included in (7)). Here, $\omega _{*}$ denotes the imaginary part of the leading eigenvalue, i.e., the eigenvalue with the largest real part. It is noted that $\kappa \rightarrow \infty$ for $u \rightarrow 0$ and, with reference to Appendix B, that the Theodorsen function $C(\kappa ) \sim {\textstyle {\frac{1}{2}}} + \mathrm {i}\, 0$ for $\kappa \rightarrow \infty$. It is seen that $\kappa$ increases approximately linearly from $\kappa \approx 0.31$ at $u = 10$ to $\kappa \approx 0.42$ at $u = 80$. At the critical flow speed $u_c = 7.05$, $\kappa \approx 0.34$.

Fig. 15

Wake wavenumber $\kappa = \omega _{*}/2 u$ as function of the flow speed u, for the case $\rho = 0.2$, corresponding to the eigenvalue curves shown in Fig. 7. Here, $\omega _{*}$ is the imaginary part of the leading eigenvalue, i.e., the eigenvalue with the largest real part $\alpha _{*}$. It is noted that $\kappa \rightarrow \infty$ for $u \rightarrow 0$

Figure 16a, b shows the real and imaginary parts of $C(\kappa )$ as function of the flow speed u for $\rho$ = 0.2, evaluated with $\kappa$ as shown in Fig. 15. Part (a) shows that ${Re}(C)= F$ decreases approximately linearly from $F \approx 0.66$ at $u = 10$ to $F \approx 0.62$ at $u = 80$. From Part (b), it is seen that $\mathrm {Im}(C)= -{\bar{G}}$ increases, also approximately linearly, from $-{\bar{G}} \approx -0.18$ at $u = 10$ to $-{\bar{G}} \approx -0.16$ at $u = 80$. At the critical flow speed $u_c = 7.05$, $F \approx 0.64$ and $-{\bar{G}} \approx -0.17$.

Fig. 16

a real and b imaginary parts of the Theodorsen function, $F = {Re}(C(\kappa ))$ and $- {\bar{G}} = \mathrm {Im}(C(\kappa ))$, respectively, as function of the flow speed u, when evaluated with the wake wavenumber $\kappa (u)$ shown in Fig. 15 ($\rho = 0.2$)

Figure 17 shows the wake wavenumber $\kappa = \omega _{*}/2 u$ as function of the flow speed u for $\rho = 25$, corresponding to the eigenvalue curves shown in Fig. 9 (again without the Coriolis and centrifugal force terms included in (7)). It is seen that $\kappa$ increases approximately linearly from $\kappa \approx 0.91$ at $u = 2.8$ to $\kappa \approx 1.1$ at $u = 12$. At the critical flow speed $u_c = 2.82$, $\kappa \approx 0.91$ as well.

Fig. 17

Wake wavenumber $\kappa = \omega _{*}/2 u$ as function of the flow speed u for the case $\rho = 25$, corresponding to the eigenvalue curves shown in Fig. 9

Fig. 18

a real and b imaginary parts of the Theodorsen function $C(\kappa )$ as function of the flow speed u, when evaluated with the wavenumber $\kappa (u)$ shown in Fig. 17 ($\rho = 25$)

Fig. 19

Wake wavenumber $\kappa = \omega _{*}/2 u$ as function of the flow speed u for the case $\rho = 0.2$, corresponding to the eigenvalue curves shown in Fig. 11

Fig. 20

a real and b imaginary parts of the Theodorsen function $C(\kappa )$ as function of the flow speed u, when evaluated with the wake wavenumber $\kappa (u)$ shown in Fig. 19 ($\rho = 0.2$)

Figures 18a, b show the real and imaginary parts of $C(\kappa )$ as function of the flow speed u, for the $\kappa$ distribution shown in Fig. 17. In the range $u \in [2.8, 12]$, these functions change approximately linearly too: ${Re}(C)= F$ decreases from 0.55 to 0.53 (Part (a)), while $\mathrm {Im}(C)= -{\bar{G}}$ increases from $-0.11$ to $-0.094$ (Part (b)). At the critical flow speed $u_c$ = 2.82, $F \approx$ 0.55 and $-{\bar{G}} \approx -0.11$.

Figure 19 shows the wake wavenumber $\kappa = \omega _{*}/2 u$ as function of the flow speed u for $\rho = 0.2$, corresponding to the eigenvalue curves shown in Fig. 11 (in this case with inclusion of the Coriolis and centrifugal force terms in (7)). Here, $\kappa$ decreases approximately linearly from $\kappa \approx 0.5$ at $u = 4$ to $\kappa \approx 0.25$ at $u = 14$. This behavior, different from the previous two cases, can be understood from Fig. 11b, which shows that the frequency parameters (the imaginary parts of the eigenvalues) here decrease with increasing flow speed. At the critical flow speed $u_c = 9.78$, $\kappa \approx 0.38$.

Figures 20a, b show the real and imaginary parts of $C(\kappa )$ as function of the flow speed u, for the $\kappa$ distribution shown in Fig. 19. As in the previous cases, in the range where $\kappa$ varies approximately linearly with u, these functions vary approximately linearly too. In the range $u \in [4, 14]$, ${Re}(C)= F$ increases from 0.60 to 0.69 (Part (a)), while $\mathrm {Im}(C)= -{\bar{G}}$ decreases from $-0.15$ to $-0.18$ (Part (b)). At the critical flow speed $u_c$ = 9.78, $F \approx$ 0.63 and $-{\bar{G}} \approx -0.17$.