# Fluid-structure interactions on a pendular 3D bluff body

Antoine Myskiw, Yann Haffner, François Paille, Jacques Borée, Christophe Sicot

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Antoine Myskiw, Yann Haffner, François Paille, Jacques Borée, Christophe Sicot. Fluid-structure interactions on a pendular 3D bluff body. ERCOFTAC Symposium "Multiphysics critical flow dynamics involving moving/ deformable structures with design applications", European Research Community on Flow, Turbulence and Combustion (ERCOFTAC), Jun 2023, Toulouse, France. hal-04715640

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Distributed under a Creative Commons Attribution 4.0 International License
FLUID-STRUCTURE INTERACTIONS ON A PENDULAR 3D
BLUFF BODY

Antoine MYSKIW¹, Yann HAFFNER²,

François PAILLÉ¹, Jacques BORÉE¹, Christophe SICOT¹

¹Institut Pprime - UPR 3346, CNRS-ENSMA-Université de Poitiers,
86360 Futuroscope-Chasseneuil, France

²Centre Scientifique et Technique du Bâtiment (CSTB), 44323 Nantes, France

**Abstract.**

The stability and the dynamics of a tridimensional bluff body pendulum are studied experimentally. The pendulum is made of a cube hung to a rod, with a bearing at the end of the rod making the system free to rotate in the cross-stream direction. High Reynolds numbers based on the side length of the cube are investigated ($Re \sim 10^5$). When the cube faces the flow, the pendular system exhibits a galloping instability resulting in the exponential growth of its swinging motion. The stability map of the system is experimentally derived with the static yaw angle of the cube as parameter, and compared with the stability map provided by a quasi-steady linear model of the motion. The stability of the pendulum appears to be very sensitive to the static yaw angle of the cube. The quasi-steady hypothesis is experimentally validated for the reduced velocities investigated but does not hold during the complete development of the swinging motion. Indeed, arising unsteady phenomena drive the saturation of the pendulum oscillations.

**Key words:** galloping instability, pendulum, 3D bluff-body, separation-reattachment mechanisms.

# 1 Introduction

Fluid-structure interactions (FSI) occurring on a bluff body is a wide subject rais-
ing issues ranging from structural safety in wind or water engineering to energy
harvesting from flows [1]. FSI on bidimensional cylindrical bodies have multiple
practical applications in civil engineering and have been extensively studied. Less
investigated, FSI between a tridimensional tethered bluff body and the surrounding
wind has nevertheless an importance in many cases, such as safety issues concerning
the loads lifted by a tower crane or in cable car engineering.

The reduced velocity $U^*$ is used as a key parameter to characterize the ratio between
fluid and solid frequencies [2, 3], which can be related to the type of FSI that may
occur. It is defined as the ratio between the advective frequency of the flow $U_0/D$
- based on the free stream velocity $U_0$ and the side length $D$ of the solid - and the
natural frequency $f$ of the solid. Two distinct mechanisms of interaction leading
to FSI may arise between a bluff body and the surrounding flow: modal coupling
such as vortex-induced vibration (VIV) and aerodynamic coupling such as flutter,
including galloping. On bidimensional bodies, VIV tend to occur at a reduced
velocity typically around 5 [2], whereas galloping occurs at large reduced velocities. The galloping instability is a common phenomenon. It can, for instance, be observed on bidimensional bluff bodies with angular shapes subjected to the wind, such as the deck of bridges in wind engineering where the DoF in the motion is usually a cross-flow oscillation or a torsional motion [4]. The shape of the bluff body is the main parameter governing the apparition of a galloping instability because of the sensitivity of the mean aerodynamic loads to the attitude of the body. The problem of a tridimensional bluff body in FSI is less commonly tackled due to its increased flow complexity [5, 6, 7, 8], and most of the existing studies focus on low Reynolds numbers.

Galloping occurs on bidimensional rectangular cylinders whose the cross section has a not too high aspect ratio - defined as the length over the cross stream thickness of the section - for which the flow does not reattach on the sides [4, 9, 10, 11]. An understanding of the basic mechanism of the galloping instability occurring on a rectangular cylinder facing the mean flow is proposed by Nakamura and al. [12] from flow visualisations and local pressure measurements. In both lateral sides of the static prism, the flow separates at the leading edges and does not reattach as the aspect ratio is low enough, creating a recirculation area from the wake with nearly flat pressure distribution along both lateral sides. A high reduced velocity ensures that a transversal motion of the body is slow enough not to produce a too large yaw angle that could cause a reattachment on the incident lateral side. Thus, during a transversal motion, the shear layer on the motion side (incident side) moves closer to the body and becomes more curved, whereas the shear layer on the opposite side moves away from it and becomes less curved. Consequently, the pressure on the sides is unbalanced with a stronger depression on the incident side, resulting in a global pressure force that amplifies the motion.

This paper tackles the characteristics of the galloping instability observed on a tethered tridimensional bluff body, a cube pendulum, at high Reynolds numbers ($Re \sim 10^5$) and high reduced velocities from 17.6 to 82.4.

# 2 Experiments and measurement methods

## 2.1 Main characteristics of the setups

Two complementary setups involving a tethered cube in two different subsonic closed-loop wind tunnels at two different scales are used. An experiment is performed inside the S120 wind tunnel at ISAE-ENSMA and another one inside the Jules Verne (JV) wind tunnel of CSTB in Nantes. In both experiments, a cube is connected by a rod to a cardan joint fixed to a rigid steel structure. The tethered cube in the JV setup is 2.5 times larger than the one in the S120 setup. The cardan joints are made of two almost perfect bearings and one bearing is blocked to allow only one DoF: the circular motion around the freestream direction (figure 1a). The mechanical damping of each bearing has been experimentally characterized and is one to two orders of magnitude smaller than the aerodynamic damping for both experiments. Table 1 introduces the main characteristics of the two setups.
<table><caption>Table 1: Main characteristics of the setups: the area A of the test section, the side length D of the cube, the length H of the rod, the mass m of the tethered system and the mechanical damping λm produced by the bearing</caption><thead><tr><th>Setup</th><th><i>A</i> (m<sup>2</sup>)</th><th><i>D</i> (m)</th><th><i>H</i> (m)</th><th><i>m</i> (kg)</th><th>λ<sub>m</sub> (s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>S120</td><td>1.2</td><td>0.2</td><td>0.2</td><td>3.3</td><td>5 × 10<sup>-3</sup></td></tr><tr><td>JV</td><td>30</td><td>0.5</td><td>0.6</td><td>19.8</td><td>1.5 × 10<sup>-3</sup></td></tr></tbody></table>

The S120 wind tunnel has an octagonal test section of 1.2m². This setup does not allow to investigate the cube swinging at too high angle amplitudes because the body gets close to the walls, so its movement might be influenced by the wall proximity [13, 14, 15]. Ramamurthy and Lee [13] showed that the influence of the wall proximity on the drag coefficient of a prism with equilateral triangle section can be neglected when the distance between them is larger than the characteristic length of the prism section. For a circular cylinder, this distance increases up to twice the cylinder diameter. Even if a tethered cube is not a prism, this information lead us not to consider the measurements when the cube makes an angle larger than 15° with the vertical axis in the S120 setup, which corresponds to a maximal distance of 2D between the edges of the body and the walls of the test section. On the opposite, the influence of the walls can be neglected in the JV setup for all angles of oscillation as the minimal distance between the walls and the swinging cube is four times larger than the side length of the cube D.

In both experiments, the blocage ratio of the tethered cube is below 5% - 3.4% and 0.8% respectively for the S120 and the JV setups - so the flow blockage is expected to have a negligible impact on the measured aerodynamic coefficients [14, 15, 16]. The flow in both wind tunnels is considered non-turbulent as the turbulence intensity is below 1%. In the S120 wind tunnel, the free-stream turbulence intensity of the flow is even below 0.3%.

## 2.2 Two complementary setups

Although they are similar, the two setups are used in different and complementary ways. The S120 setup is used to study the sensitivity of the stability of the system to the static yaw angle β₀ (figure 1b) of the cube at a constant free stream velocity U₀. This angle, illustrated in figure 1a, corresponds to the angular position of the cube around the rod that can be set before any experiment: the cube cannot rotate around the rod during the motion. In this setup, the experimental dynamics of the system is confronted to a theoretical quasi-static model built to predict the stability in a smooth or turbulent flow.

In the JV setup, the angle β₀ is blocked at 0°, so the cube faces the mean flow that is non-turbulent. In this configuration, a galloping instability occurs and is investigated with pressure probes at different flow speeds. The reduced velocity U* is an important parameter of this setup. It represents the ratio between the period of the swinging motion 1/f – here expressed from the natural frequency of the swinging motion f – and the convective time-scale of the flow D/U₀, U₀ being the mean flow velocity. The pendulum frequency f is expressed from the distance L between the centre of gravity of the system to the fixed point. In both setups,
the rod is light enough compared to the cube so that the centre of gravity of the whole system can be approximated as the barycentre of the cube : $f = \sqrt{g/L}$ and $L = H + D/2$.

A single high reduced velocity is studied in the S120 setup, whereas a wide range of lower reduced velocities is investigated in the JV setup. Indeed, the larger scale of the JV setup makes it possible to investigate lower reduced velocities. Table 2 summarizes the ranges of the parameters investigated in each setup.

<table><caption>Table 2: Parameters investigated in the two setups: static yaw angle $β_0$, mean flow velocity $U_0$, reduced velocity $U^*$ and Reynolds number $Re$</caption><thead><tr><th>Setup</th><th>β₀ (°)</th><th>U₀ (m/s)</th><th>U*</th><th>Re</th></tr></thead><tbody><tr><td>S120</td><td>[0, 45]</td><td>15</td><td>82.4</td><td>1.9 × 10⁵</td></tr><tr><td>JV</td><td>0</td><td>[4.8, 14.6]</td><td>[17.6, 54]</td><td>[1.5, 4.7] × 10⁵</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>Figure 1: (a, b) Conventions used in the study: definition of the angle made by the cabin $φ$, its static yaw angle $β_0$ and its dynamic yaw angle $δβ$</figcaption></figure>

(c) Integration of the aerodynamic loads from the pressure probes corona over the lateral, front and rear faces of the cube

## 2.3 Metrology

In both setups, a motion tracking by cameras is realized on the cube. In the S120 setup, 4 cameras Vicon V5 5Mpx with a focal length of 12.5mm track 4 circular markers of diameter 0.01D on the rear face of the cube at a rate of 200Hz. In the JV setup, 5 cameras Qualisys Oqus 4Mpx with a focal length of 20mm track 4 circular markers of diameter 0.02D on the bottom face of the cube at a rate of 200Hz. The motion tracking of the JV setup is complemented by a set of 2 unidirectional accelerometers Kistler 8316A with a range of ±10g sensitive to both static and dynamic movements. It provides the angles of the pendulum to the vertical axis with a precision of ±0.5°.

In the S120 setup, a strain gauge 6-component aerodynamic balance is used to provide the static aerodynamic loads on the cube with a maximal incertitude of ±5% at the reference speed $U_0 = 15m/s$. The balance is mounted inside a profiled arm lifting the cube from the ground of the wind tunnel. In the JV setup, the static loads on the tethered cube are measured using an ATI Omega 170 strain gauge
6-component aerodynamic balance, mounted at the base of the steel structure over the cardan joint, with a precision of ±3% on the measured loads for the reference configuration at $U_0 = 15\,m/s$. In both setup, the aerodynamic loads on the cube are measured with the balance at static angles of attack between 0° and 45°. This range can then be extended to [−180°, 180°] by symmetry considerations.

The whole arrangement of the suspended cube and the steel structure of the JV setup has been dynamically characterized in both torsion and flexion to ensure that its structural behaviour would not interfere with nor with the measurements, neither in the fluid-structure coupling. The main modes of the structure appear decoupled from the pendulum dynamics as the lowest one is already 3 times higher than the natural frequency of the pendulum $f \sim 0.5\,Hz$.

Finally, local unsteady pressure measurements are performed with a corona of 32 pressure taps on the front, rear and lateral faces of the JV cube as depicted in figure 1c. These pressure taps are linked to two pressure scanners PSI 32HD of ±2500 Pa range by 1m long vinyl tubings. The measurements are a posteriori corrected to compensate for the pressure distortions induced by the tubings using a priory calibration of the tubing system in an in-house test bench. The frequency response of the system is flat on the whole range of measurement up to 200Hz. For the reference configuration in a mean flow at $U_0 = 15\,m/s$, measurements have an uncertainty of ±5% of the reference dynamic pressure. Reference conditions (static pressure, temperature and air humidity) are taken in test section where the cube is located. The reference static and dynamic pressures are measured at a distance $5D = 2.5\,m$ upstream to ensure homogeneous flow conditions.

The local pressure is used to compute the aerodynamic loads by integration over the cube faces. On the lateral, rear and front faces, each local pressure is integrated over a vertical strip like on the scheme in figure 1c, and summed up to compute the global pressure loads acting on the face. The aerodynamic loads are then derived from the pressure loads on the faces. The static loads on the cube obtained with this method match well with the static loads measured with the aerodynamic balance, with a maximal difference of 4.5% relative to the mean drag coefficient measured with the balance. This observation for the 3D cubic geometry is very important because this methodology provides a straightforward way to determine experimentally the dynamic loads on the swinging cubes and to discuss the departure from quasi-steady flow conditions.

### 3 Stability analysis: a quasi-steady model of the bluff body pendulum dynamics

The stability of the pendulum system is assessed using the quasi-steady hypothesis to use the mean static drag and lateral lift coefficients $C_D$ and $C_L$ of the tethered bluff body in the pendulum momentum equation. The linearization of these equations for a small angular displacement from the equilibrium position $\varepsilon = \phi - \bar{\phi}$ ($\phi$ being the angular position of the pendulum as illustrated in figure 1a) gives the following second order differential equation in $\varepsilon$ (1) with the pseudo-frequency $\omega_0$ (2) and the aerodynamic damping coefficient $\lambda$ (3) of the linear model. $g = 9.81\,m.s^{-2}$ represents the gravitational acceleration, $\rho$ the air density and $S$ the reference cube surface. $\lambda'$ (4) represents the reduced aerodynamic damping coefficient depending only on the
static mean aerodynamic characteristics of the studied bluff body. If the angular position at equilibrium $\phi(\beta_0) = 0$, $\omega_0$ is the pseudo-frequency of a free pendulum $\sqrt{g/L}$.

$$ \ddot{\epsilon} + 2\lambda \dot{\epsilon} + \omega_0^2 \epsilon = 0 \qquad (1) $$

$$ \omega_0 = \sqrt{\frac{g}{L} \cos(\bar{\phi}(\beta_0))} \qquad (2) $$

$$ \lambda = \frac{\rho S U_0}{4m} \lambda' \qquad (3) $$

$$ \lambda' = \bar{C}_D(\beta_0) + \frac{d\bar{C}_L}{d\beta} \bigg|_{\beta_0} \qquad (4) $$

Thus, the pendulum system is theoretically stable if $\lambda > 0$ and unstable if $\lambda < 0$. As the sign of $\lambda$ is driven by the reduced damping coefficient $\lambda'$, the system is unstable if $\bar{C}_D(\beta_0) + \frac{d\bar{C}_L}{d\beta} \bigg|_{\beta_0} < 0$. This inequation corresponds to the Den Hartog criterion, characteristic of the galloping instability [17], showing that the transverse galloping instability is characterized by a negative static lift gradient that counterbalance the stabilizing drag. The static mean aerodynamic coefficients of the cube are measured to draw the stability map of the tethered cube with the static yaw angle $\beta_0$ as parameter for $U_0 = 15m/s$ (figure 2a) and are in line with the data gathered from the literature on a prism with a square cross section in a non-turbulent flow [4, 10, 18, 19, 20]. The pendulum is theoretically unstable when the cube faces or nearly faces the flow with an angle $\beta_0 \in [-9^\circ, 9^\circ]$.

<figure><img src="image_3.png" /><figcaption>Figure 2: a) Theoretical reduced aerodynamic damping coefficient $\lambda'(\beta_0)$ of the tethered cube in a non-turbulent flow plotted with the static mean aerodynamic coefficients of the cube $\bar{C}_D$ (drag), $\bar{C}_L$ (lateral aerodynamic force) and $\bar{C}_\phi = \bar{C}_L \cos(\beta_0) - \bar{C}_D \sin(\beta_0)$ (along the axis of the motion $\vec{e}_\phi$) measured for $U_0 = 15m/s$ ($U^* = 82.4$) ; the red area corresponds to the theoretical unstable region with $\lambda'(\beta_0) < 0$</figcaption></figure>

b) Standard deviation $\sigma_\phi$ of the angle $\phi$ derived from measurements at $U_0 = 15m/s$ for different static yaw angles $\beta_0$ ; the red area shows the approximate unstable regions

Some dynamic experimental observations are realised in the same configurations (same body, same flow characteristics) to assess the linear model for different static yaw angles $\beta_0$ from 0° to 40°. This corresponds to a high reduced velocity $U^* = 82.4$
for which the quasi-steady theory is expected to be justified. The standard deviation of the pendulum angular position $\sigma_{\phi}$ is derived from dynamic measurements at each $\beta_0$ investigated to visualize the range for which galloping occurs. $\sigma_{\phi}$ is obviously much higher in an unstable configuration than in a stable one, but its exact value for unstable configurations is not relevant as the amplitude of the swinging motion grows exponentially and the experiment has to be stopped before reaching a too large amplitude. The results are presented in figure 2b. : the experimental unstable region is approximately the same as the theoretical one.

These results validate the linear quasi-steady model as a way to predict the stability of a bluff body pendulum from its static mean aerodynamic coefficients. The experiments realised in the JV setup focus on the galloping instability occurring when the cube faces the mean flow ($\beta_0 = 0$). As a result, the yaw angle $\beta$ measured in this setup equals the dynamic yaw angle $\delta\beta$. The next sections focus on this setup for which several reduced velocities are investigated.

## 4 The two phases of the swinging motion

Scaling the time with the theoretical damping coefficient $\lambda$ (4) introduced in the stability analysis ($t^* = -\lambda t$) reveals a quasi-superimposed evolution of the envelope of the oscillations of the yaw angle $E_{\beta}$ for the reduced velocities $U^* = 37.7$ and, $U^* = 54$ (figure 3a). This similitude shows that the development of the swinging motion induced by galloping is mainly driven by the yaw angle $\beta$, and more accurately the absolute envelope of the yaw angle $|E_{\beta}|$. This angle can be expressed with the absolute envelope of the pendulum oscillations $|E_{\phi}|$ and the reduced velocity $U^*$ (f is the oscillation frequency, and $|E_{\phi}| = 2\pi f |E_{\phi}|$):

$$|E_{\beta}| \approx \tan(|E_{\beta}|) = \frac{L|E_{\phi}|}{U_0} = \frac{2\pi f L|E_{\phi}|}{U_0} = 2\pi \frac{L|E_{\phi}|}{D U^*} \quad (5)$$

From equation (5), knowing that $L/D$ is constant in the experiments, the similarity in $|E_{\beta}|$ means that the higher the reduced velocity, the larger the amplitude of the pendulum oscillations. For the two reduced velocities 37.7 and 54, the experiments had to be stopped because the setup could not support too large amplitudes. However, the beginning of a saturation of the oscillations is visible for $U^* = 37.7$ after the quasi-steady exponential amplification that could not be observed entirely (figure 3a). Thus, to observe the full development of the swinging motion induced by galloping and its saturation, lower reduced velocities down to 17.6 are investigated by reducing the flow velocity (figure 3b). Two phases can be identified from the evolution of the motion and the evolution of the energy received by the system from the flow. The first one is a quasi-exponential amplification of the amplitude of the oscillations that ends when $|E_{\beta}|$ reaches approximately 6°. The saturation then begins and the growth rate of the amplitude of the oscillation decreases down to zero, leading to a saturated state with $|E_{\beta}| \approx 12.5°$. This angle seems independent of the reduced velocity, as shown in figure 3b.
<figure><img src="image_4.png" /><figcaption>Figure 3: Envelope $E_β$ of the oscillations of the yaw angle $β$ during a galloping instability:</figcaption></figure>

a) at $U^* = 37.7$ and $U^* = 54$ according to the reduced time $t^* = -\lambda t$, time scaled by the theoretical damping coefficient $\lambda$

b) at 3 successive low reduced velocities $U^*$ to investigate the saturated state; the JV wind tunnel velocity $U_0$ is increased step by step in this dedicated experiment

# 5 Detailed analysis of the swinging motion

## 5.1 A quasi-steady amplification phase

The quasi-exponential amplification of the amplitude of oscillation is studied from the local pressure measurements. Unbalanced pressure loads on the lateral faces tend to amplify the motion, as presented in figure 4 for $U^* = 54$ at $β = ±3.8°$. This is similar to the observations of Nakamura and al. [12] on galloping cylinders with a square cross section. Thus, it seems that the mechanism of galloping occurring on the cube pendulum and on a bidimensional cylinder with a square section are the same despite the tridimensional shape of the studied cubic body.

<figure><img src="image_5.png" /><figcaption>Figure 4: Visualisation of local pressure coefficients $C_p$ over the cube faces measured during the amplification phase of the swinging motion at $U^* = 54$</figcaption></figure>
The local pressure measurements are integrated to get the instantaneous aerodynamic loads acting on the cube during its swinging motion. A phase-averaging of these aerodynamic loads over several period of oscillation is then used to get the evolution of the dynamic loads with the yaw angle $\beta$. Figure 5 presents the evolution of the dynamic and static aerodynamic lateral coefficients $C_{\phi}(\beta)$. The dynamic loads in the two half-periods of the swinging motion ($\phi > 0$ and $\phi < 0$) fit well with the static loads for the two reduced velocities. The negative slope of the aerodynamic force - characteristic of galloping - is well visible on both static and dynamic loads evolution. Consequently, the behaviour of the system is quasi-steady, and the stability analysis using the quasi-steady theory is justified.

<figure><img src="image_6.png" /><figcaption>Figure 5: Dynamic and static coefficient $C_{\phi}$ of the cube derived from the phase-averaged local pressure measurements according to the yaw angle $\beta$ during the amplification phase, $U^* = 37.7$ and $U^* = 54$</figcaption></figure>

## 5.2 An unsteady saturation

The same analysis based on the dynamic loads obtained from the pressure measurements is realised for the saturation phase of the motion for $U^* = 17.6$ and $U^* = 37.7$. The dynamic loads of the two half-periods of oscillation ($\phi > 0$ and $\phi < 0$) dissociate to form a cycle, as shown in figure 6. The cycle gets larger and larger until the saturated state is reached for $U^* = 17.6$, but not for $U^* = 37.7$ as the experiment had to be stopped not to reach the mechanical limits of the setup in amplitude. The pendulum dynamics is thus driven by unsteady phenomena, like an airfoil subjected to dynamic stall [21]. This starts after $E_{\beta}$ reaches approximately $6^{\circ}$, which corresponds to the end of the quasi-exponential amplification.

The power balance between the cube pendulum and the flow can be investigated as the power received by the pendulum is simply the product of its speed and the aerodynamic loads. During the quasi-steady amplification, the pendulum only receives energy from the flow, whereas in saturated state, the pendulum gives energy to the flow while going down and gets energy while going up (figure 6). When the motion is saturated, the periodic energy exchanges are balanced.
Finally, the angle $|E_β| \approx 12.5°$ reached in saturated state is very close to the 13° characteristic of the flow reattachment on a static cylinder with a square cross section [4, 19, 20, 22]. Thus, the origin of this saturation and the energy exchanges described above might be related to flow separation-reattachment mechanisms. This issue is not further investigated in this paper.

<figure><img src="image_7.png" /><figcaption>Figure 6: Dynamic coefficient $C_φ$ of the cube derived from the phase-averaged local pressure measurements according to the yaw angle $β$ during the saturation phase, for the low reduced velocities $U^* \approx 17.6$ and $37.7$; P corresponds to the power received by the cube pendulum from the flow</figcaption></figure>

# 6 Conclusions

A galloping instability is observed for a cube pendulum subjected to a transverse non-turbulent flow at high Reynolds number and for a wide range of reduced velocities. It occurs when the cube makes a static yaw angle $β_0 \in [-9°, 9°]$ with the mean flow. The quasi-steady model based on the linearized momentum equations leads to the same Den Hartog criterion as for galloping bidimensional cylinders, and proves to be a relevant tool to predict the stability of a tethered cube. Experimental measurements of the dynamic loads on the cube validate the quasi-steady theory. Despite the tridimensional shape of the cube pendulum, the galloping instability observed seems to be driven by the same mechanisms as the galloping instability observed on bidimensional cylinders with square cross sections.
The study of the full swinging motion at $\beta_0 = 0°$ reveals an unsteady aerodynamic behaviour of the cube pendulum starting when the envelope of the yaw angle $E_\beta$ reaches $\pm 6°$. This marks the beginning of a saturation of the oscillations up to $|E_\beta| \approx 12.5°$, value that seems independent of the flow velocity and probably related to the shape of the bluff body and its aerodynamic behaviour with a yaw angle. The understanding of this phenomenon is in progress.

## Acknowledgements

The PhD of Antoine Myskiw is supported by the ANR Project TurbCab (ANR-22-CE22-0010). All the data bases used in this paper were obtained before the start of project TurbCab.

## References

[1] Mehdipour I., Madaro F., Rizzi F., De Vittorio M. *Comprehensive experimental study on bluff body shapes for vortex-induced vibration piezoelectric energy harvesting mechanisms*. Energy Conversion and Management: X, 13, 2022.

[2] Blevins R. D. *Flow-induced vibration, 2nd edition*. Krieger Publishing Company, 2001.

[3] Hémon P. *Vibrations des structures couplées avec le vent*. École Polytechnique, 2006.

[4] Parkinson G. V. *Wind-induced instability of structures*. Philosophical Transactions of the Royal society of London, 269:395-409, 1971.

[5] Kozmar H. *Flow, turbulence and surface pressure on a wall-mounted cube in turbulent boundary layers*. J. Wind Engineering and Industrial Aerodynamics, 210, 2021.

[6] Zhao J., Sheridan J., Hourigan K., Thompson M. C. *Flow-induced vibration of a cube orientated at different incidence angles*. J. Fluids and Structures 91, 2019.

[7] Khan M. H., Sooraj P., Sharma A., Agrawal A. *Flow around a cube for Reynolds numbers between 500 and 55000*. Experimental Thermal and Fluid Science, 93:257-271, 2018.

[8] Govardhan R. N., Williamson C. H. K. *Vortex-induced vibrations of a sphere*. J. Fluid Mechanics, 531:11-47, 2005.

[9] Parkinson G. V., Brooks N. P. H. *On the aeroelastic instability of bluff cylinders*. J. Applied Mechanics, 28:252-258, 1961.

[10] Smith J. D. *An experimental study of aeroelastic instability of rectangular cylinders*. M.A.Sc. Thesis, University of British Columbia, 1962.

[11] Nakamura Y., Tomonari Y. *Galloping of rectangular prims in a smooth and in a turbulent flow*. J. Sound and Vibration, 52:233-241, 1977.

[12] Nakamura Y., Hirata K., Urabe T. *Galloping of rectangular cylinder in the presence of a splitter plate*. J. Fluids and Structures, 5:521-549, 1991.

[13] Ramamurthy A. S., Lee P. M. *Wall effects on flow past bluff bodies*. J. Sound and Vibration, 31:443-451, 1973.

[14] Barlow J. B., Rae W. H., Pope A. *Low speed wind tunnel testing*. John Wiley & Sons, 1999.
[15] Sharma S., Raghav V., Komerath N., and Smith M. *Wall effect on fluid-structure interactions of a tethered bluff body*. Physics Letters A, 377:2079-2082, 2013.

[16] West J. S., Apelt C. J. *The effect of tunnel blockage and aspect ratio on the mean flow past a circular cylinder with Reynolds numbers between 10⁴ and 10⁵*. J. Fluid Mechanics, 114:361-377, 1982.

[17] Den Hartog J. *Mechanical Vibrations, 4th ed*. Dover, New York, 1984.

[18] Parkinson G. V., Smith J. D. *The square prism as an aeroelastic non-linear oscillator*. Q. J. Mechanics and Applied Mathematics, 17:225-239, 1964.

[19] Igarashi T. *Characteristics of the flow around a square prism*. Bulletin of JSME, 27, 1984.

[20] Bearman P. W., Gartshore I. S., Maull D. J., Parkinson G. V. *Experiments on flow-induced vibration of a square-section cylinder*. J. Fluids and Structures, 1:19-34, 1987.

[21] McCroskey W. J. *The phenomenon of dynamic stall*. NASA Ames Research Center Moffett Field, CA, United States, 1981.

[22] Robertson J. M., Wedding J. B., Peterka J. A., Cermak J. E. *Wall pressures of separation-reattachment flow on a square prism in uniform flow*. J. Industrial Aerodynamics, 2:345-359, 1977.

[23] Nakamura Y., Hirata K. *Pressure fluctuations on oscillating rectangular cylinders with the long side normal to the flow*. J. Fluids and Structures, 5:165-183, 1991.

[24] Matsumoto M. *Vortex shedding of bluff bodies: a review*. J. Fluids and Structures, 13:791-811, 1999.

[25] Parkinson G. V., Wawzonek H. A. *Some considerations on of combined effects of galloping and vortex resonance*. J. Wind Engineering and Industrial Aerodynamics, 8:135-143, 1981.

[26] Williamson C. H. K., Govardhan R. *Vortex-induced vibrations*. Annual Review of Fluid Mechanics, 36:413-455, 2004.

[27] Zhao J., Leontini J. S., Lo Jacono D., Sheridan J. *Fluid-structure interaction of a square cylinder at different angles of attack*. J. Fluid Mechanics, 747:688-721, 2014.