# Linear Stability Analysis of Two Parallel Inverted Flags in Axial Flow

Shaoguang Wang, Mathias Legrand, Michael P Païdoussis

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Shaoguang Wang, Mathias Legrand, Michael P Païdoussis. Linear Stability Analysis of Two Parallel Inverted Flags in Axial Flow. 7th Symposium on Fluid-Structure-Sound Interactions and Control, Jul 2025, Toronto, Canada. [hal-05206840](https://hal.science/hal-05206840v1)

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# Linear Stability Analysis of Two Parallel Inverted Flags in Axial Flow

Shaoguang Wang, Mathias Legrand, and Michael P. Païdoussis

Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada

**Abstract.** The dynamics of two identical parallel thin flexible plates with free leading edges and clamped trailing edges, referred to as “inverted flags”, is investigated theoretically. A linear model is developed by adopting a number of assumptions: (1) the Euler-Bernoulli beam theory is applied for both flags, and (2) incompressible potential flow theory is used for the fluid model. The space-dependence of the potentials is reduced through a Fourier transform, and the pressure jumps across the flags are expressed in terms of the flag displacements as integral equations involving Cauchy principal values. The flag displacements are approximated as superpositions of eigenfunctions of a cantilever beam in vacuum, and the motion of the flags is obtained through the Galerkin method. The main purpose of the model is to predict the critical flow velocity at which the onset of instability of the system occurs and the sensitivity of that critical flow velocity to the gap separating the two flags. It is found that the critical flow velocity becomes independent of the flag gap-to-length ratio when it is approximately greater than 1. When the ratio is below unity, the critical flow velocity reduces with the ratio.

**Keywords:** Inverted Flags, Linear Stability Analysis.

## 1 Introduction

Inverted flags have been of great interest in energy harvesting systems due to large-amplitude flapping they might feature and large strain energy they can store [1]. An underwater energy harvester can start up at extremely low current speeds and have a wide working range [2]. Inverted flags are also considered in order to enhance the performance of some types of heat exchangers [3]. Flag-fluid models are even applied in biological systems [4]; for example, heart valve leaflet interaction with blood flow is a typical fluid-structure interaction [5].

Single as well as multiple inverted flags have been reported to present large-amplitude flapping above a critical flow velocity [6, 7]. Recent numerical and experimental investigations have focused on inverted-flags-energy-harvesting-system in a pipe flow [8], in oscillating flows [9], and in uniform shear flows [10]. Extensions to various flag arrangements are also reported and suggest that the side-by-side arrangement is more favourable to energy harvesting [11].

The problem of interest is sketched in Fig. 1. The aim of the present study is to compare the critical flow velocities of two parallel flags using the Fourier transform method with other existing numerical and experimental results. The effects of the gap between the two flags on the critical flow velocity is also of interest.
<figure><img src="image_2.png" /><figcaption>Figure 1: Two parallel inverted flags in axial flow</figcaption></figure>

## 2 Theoretical Model

The equation of motion of the two flags is derived from Euler-Bernoulli beam theory. For flag j ($j = 1, 2$), the transverse displacement $w_j(x, t)$, $\forall x \in (0, L)$ reads

$$ \rho_p h_p \ddot{w}_j + D \partial_x^4 w_j + p_j = 0, \quad (1) $$

where $\rho_p$ is the flag density, $h_p$, the flag thickness, $D = Eh_p^3/[12(1 - \nu^2)]$, the flag stiffness, $E$, Young's modulus, $\nu$, Poisson's ratio, and $p_j(x, t)$, the fluid pressure difference across flag j. The boundary conditions of the flags follow:

$$ w_j(0, t) = \partial_x w_j(0, t) = \partial_x^2 w_j(L, t) = 0, \quad D \partial_x^3 w_j(L, t) = \rho_f L U_\infty (\dot{w}_j(L, t) - U_\infty \partial_x w_j(L, t)) \quad (2) $$

where $L$ is the flag length, $\rho_f$, the fluid density, and $U_\infty$, the free flow velocity. Note that the second boundary condition at $x = L$ involves an inviscid fluid dynamic force and the compressive force due to the frontal pressure at the upstream [12]. Incorporating the shear force boundary condition into Eq. (1) by means of a Dirac function yields

$$ \rho_p h_p \ddot{w}_j + D \partial_x^4 w_j + p_j + \rho_f L U_\infty (\dot{w}_j - U_\infty \partial_x w_j) \delta(x - L) = 0. \quad (3) $$

The system can then be analyzed with the flag treated as a standard cantilever beam subjected to simple free-clamped boundary conditions $w_j(0, t) = \partial_x w_j(0, t) = \partial_x^2 w_j(L, t) = \partial_x^3 w_j(L, t) = 0$.

The perturbation potential $\phi_n(x, y, t)$ in fluid domain $n$ ($n = 0, 1, 2$), see Fig. 1, satisfies the Laplace equation $\partial_x^2\phi_n + \partial_y^2\phi_n = \Delta\phi_n = 0$. The perturbation potential vanishes at infinity; i.e. $\phi_n(\pm\infty, y, t) = 0, \forall y \in (-\infty, \infty)$ with $n = 0, 1, 2$ and $\phi_0(x, -\infty, t) = \phi_2(x, \infty, t) = 0, \forall x \in (-\infty, \infty)$. On flag j occupying domain $\Gamma_j = \{x \in (0, L), y = (j-1)d, j = 1, 2, d$ being the gap distance between the two flags, the perturbation potential satisfies the no-penetration boundary condition $\partial_y\phi_{j-1} = \partial_y\phi_j = (\partial_t - U_\infty\partial_x)w_j$, and the pressure difference is given by the linearized unsteady Bernoulli equation $p_j = -\rho_f(\partial_t - U_\infty\partial_x)[\phi_j - \phi_{j-1}]$. Anywhere else on the boundary of a fluid domain, $p_j = 0$.

To make these equations dimensionless and analyze the problem in the frequency domain, we define the following dimensionless quantities with asterisks: $x = x^*L, y = y^*L, d = d^*L, m^* = \rho_f L/(\rho_p h_p)$, $t = t^*L^2 \sqrt{\rho_p h_p/D}$, $\omega = \sqrt{D\omega^*}/(L^2\sqrt{\rho_p h_p})$, $U_\infty = U^*\sqrt{D}/(L\sqrt{\rho_f L})$, $w_j(x, t) = w_j^*(x^*)Le^{i\omega t}$, $p_j(x, t) = p_j^*(x^*)e^{i\omega t}D/L^3$, and $\phi_n(x, y, t) = \phi_n^*(x^*, y^*)e^{i\omega t}\sqrt{D}/\sqrt{\rho_p h_p}$. Asterisks are dropped for simplicity in the remainder.

## 3 Solution Methods

The pressure differences in Eq. (1) are expressed in terms of the flag displacements by following the work on two parallel conventional (i.e., clamped-free) flags [13]

$$ \frac{1}{2\pi m} \left( \int_0^1 \frac{p'_j(\xi)}{x-\xi} d\xi + \int_0^1 \frac{x-\xi}{(x-\xi)^2 + d^2} p'_k(\xi) d\xi \right) = -\omega^2 w_j(x) - 2i\omega U w'_j(x) + U^2 w''_j(x) \quad (4) $$
considered for both pairs $(j, k) = (1, 2)$ and $(j, k) = (2, 1)$. The first integral in Eq. (4) is treated as the Cauchy principal value. It is computed numerically using the Gauss-Chebyshev quadrature scheme [14]. Due to the linearity of Eq. (4) in terms of the flag displacement, it is convenient to express the pressure difference in the form

$$p_j(x) = -\omega^2 p_j^M(x) - 2i\omega U p_j^C(x) + U^2 p_j^K(x), \quad j = 1, 2 \qquad (5)$$

where the components with superscripts M, C and K correspond to added mass, gyroscopic effects and added stiffness, respectively [15].

The Galerkin method is used to solve (1). The displacement of each flag is decomposed using the first Q eigenfunctions of a cantilever beam in vacuum

$$w_j(x) = \sum_{n=1}^{Q} A_{jn} \psi_n(x). \qquad (6)$$

Applying the Galerkin procedure in Eq. (1), one obtains

$$[-\omega^2(\mathbf{M} + \mathbf{F}_M) + i\omega\mathbf{C} + (\mathbf{K} + \mathbf{F}_K)]\mathbf{q} = \mathbf{0}, \qquad (7)$$

where M is an identity matrix with size of $2Q$, K is diagonal with $K_{(j-1)Q+i} = \lambda_i^4$ for $j = 1, 2$ and $1 \le i \le Q$ with $\lambda_i$ are the ascending roots of the cantilever beam transcendental equation $1 + \cos \lambda \cosh \lambda = 0$, and $\mathbf{q} = \{A_{11}, A_{12}, \dots, A_{1Q}, A_{21}, A_{22}, \dots, A_{2Q}\}^T$. The entries of $\mathbf{F}_M$, $\mathbf{C}$, and $\mathbf{F}_K$ read

$$F_{M,ij} = m \int_{0}^{1} P_{i}^{M} \Psi_{j} dx, \quad C_{ij} = -2mU \int_{0}^{1} P_{i}^{C} \Psi_{j} dx + \sqrt{mU} \Psi_{i}(1) \Psi_{j}(1), \qquad (8)$$

$$F_{K,ij} = mU^2 \int_{0}^{1} P_{i}^{K} \Psi_{j} dx - U^{2} \Psi_{i}^{\prime}(1) \Psi_{j}(1), \quad i, j = 1, \dots, 2Q,$$

where $\Psi = \{\psi_1, \dots, \psi_Q, \psi_1, \dots, \psi_Q\}$ and $\mathbf{P} = \{p_1, \dots, p_{2Q}\}$. The elements $p_1$ through $p_Q$ and $p_{Q+1}$ through $p_{2Q}$ are pressure solutions for flag 1 and 2 corresponding to the eigenfunctions. The superscripts M, C, and K are pressure components referring to Eq. (5).

# 4 Results and Discussion

In order to have non-trivial solutions for $\mathbf{q}$ in Eq. (7), the determinant of $[-\omega^2(\mathbf{M} + \mathbf{F}_M) + i\omega\mathbf{C} + (\mathbf{K} + \mathbf{F}_K)]$ should vanish, which yields a $4Q$-th order polynomial, the roots of which being the eigenvalues. More specifically, the eigenvalues are $2Q$ complex conjugate pairs. One set of $Q$ pairs corresponds to modes where flags are out-of-phase (the flag deflections are symmetric), and the other set, to in-phase modes (the flag deflections have the same direction). The imaginary and real parts of an eigenvalue $\Lambda$ are the frequency and damping of the corresponding mode, respectively. If the real part is positive, the mode becomes unstable: the corresponding flow velocity is the critical flow velocity $U_{cr}$.

In order to compare with published results [7], a base case with $m = 0.3$ and $d = 2/3$ is investigated first. It is found that the first mode, very similar to the first mode of a cantilever beam in vacuum, loses stability first when increasing the flow velocity, as shown in Fig. 2. The higher modes are not of interest because they lose stability at much higher flow velocities.

As the flow velocity increases, the imaginary part of the eigenvalue decreases till it becomes zero, and the real part $\Re(\Lambda)$ crosses the horizontal axis at $U_{cr} \approx 1.44$. At this point, $\Im(\Lambda) = 0$ and the first mode is subjected to static divergence instability, that is buckling. In [7], the corresponding numerical and experimental critical flow velocities are 1.53 and 1.85, respectively. Our results match better with their numerical results.
<figure><img src="image_3.png" /><figcaption>Figure 2: Stability analysis for the first mode with $m = 0.3$ and $d = 2/3$. in terms of flow velocity $U$.</figcaption></figure>

<table><caption>Table 1: Comparison of critical flow velocities by varying the gap $d$.</caption><thead><tr><th>d</th><th>1/3</th><th>2/3</th><th>1</th><th>4/3</th></tr></thead><tbody><tr><td>Present study</td><td>1.41</td><td>1.44</td><td>1.47</td><td>1.48</td></tr><tr><td>Kim & Kim [7]</td><td>1.24</td><td>1.53</td><td>1.67</td><td>1.74</td></tr><tr><td>Difference (%)</td><td>13.7</td><td>5.9</td><td>12.0</td><td>14.9</td></tr></tbody></table>

In Table 1, the sensitivity of the critical flow velocity to the gap $d$ is listed and compared with the results in [7]. Four different gaps are tested, and the largest relative difference is no more than 15%, which is considered to be reasonable. As the gap increases, the critical flow velocity also increases. It is suggested that the critical flow velocity would approach an asymptotic limit value.

Second, Kim and Kim [7] verified the independence of the mass ratio to the critical flow velocity by running the same flag configuration ($d = 1$) in a wind tunnel with mass ratio 0.3 and a water tunnel with mass ratio 250. They found that the critical flow velocities are very close, even though the mass ratios have a three-order-of-magnitude difference. Our mathematical model predicts first mode buckling at $U_{cr} \approx 1.47$ for a low mass ratio as shown in Fig. 3a, which has a small difference than the experimental value of 1.89 [7]. For a high mass ratio as shown in Fig. 3b, the model predicts flutter at near-zero flow velocity prior to the buckling, and the critical flow velocity for the buckling $U_{cr} \approx 1.95$, which matches well with the experimental value of 1.82 [7]. The flutter prediction matches the observation in the experiments reported in [12], in which they observed flutter or flutter-like oscillation at very low flow velocities and then buckling at higher flow velocities.

<figure><img src="image_4.png" /><figcaption>Figure 3: Stability analysis for low and high mass ratios for the first mode.</figcaption></figure>
# 5 Conclusion

The linear stability analysis of two parallel inverted flags with varying separating gap is investigated. The mathematical model predicts buckling and the critical flow velocities agree well with existing results [7] for the buckling instability. It is found that when the gap separating the flags increases, the critical flow velocities increase yet slowly. The two flags eventually become decoupled for very large gaps, and the critical flow velocity is similar to that for a single inverted flag. For high mass ratios, i.e. light flag or heavy fluid, the proposed mathematical model predicts flutter at near-zero flow velocity and buckling at higher flow velocity, which agrees well with the observations in the experiments [12].

**Supplementary Material** The Python scripts for the base case computation in the present study is available at the Permalink FSSIC2025_source_code.

**Acknowledgments** The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.

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