There is no critical mass ratio for galloping of a square cylinder under flow


Flow-induced vibration (FIV) is a typical fluid-structure interaction phenomenon. Although FIV has significant threats for structural safety, it also provides a promising means for harnessing energy from the flows. Generally, the FIV of square cylinders under a flow is as a result of the two interaction mechanisms: galloping and vortex-induced vibration (VIV). VIV is a consequence of the interaction between the shed vortices of the bluffy body and structural motion. In most VIV cases, the vibrations are limited as the increase of flow velocity, and the oscillation frequency follows the Strouhal law.

Numerous models have been developed to study VIV by representing its solid dynamics and unsteady fluid characteristics. In particular, reduced-order models (ROMs), allow capturing of qualitative and quantitative features of VIV. Although the ROM model has been mostly applied and validated for cylindrical structures for VIV, there are also studies on its applications to square sections for coupled VIV-galloping motion. Galloping, on the other hand, is due to the coupling between the motion and mean flow of the bluff body. Contrary to VIV, galloping is not limited to a certain velocity range, and the motion frequency is usually related to solid mode frequency irrespective of the flow velocity.

Unlike cylinders with a circular cross-section, square cylinders at zero angles are more susceptible to galloping and are thus applied as generic configuration. Square geometric also experiences VIV, resulting in combined VIV-galloping motion. A particular behavior of the amplitude reduction has been reported in light square structures with a low mass ratio between the fluid and the solid. Despite the many studies try to provide more insights into this special phenomenon, they have been limited in terms of the dimensionless flow velocity due to high computational costs. At present, there are doubts on the existence of a critical mass ratio. In addition, considering that damping affects the galloping-VIV interactions, it is imperative to understand the effects of damping on the special low mass ratio behaviors. This is of practical importance in harvesting energy from vibrations.

On this account, Dr. Peng Han from Northwestern Polytechnical University / Ecole Polytechnique, Institut Polytechnique de Paris together with Professor Emmanuel de Langre from Ecole Polytechnique, Institut Polytechnique de Paris studied the galloping and VIV of a square cylinder under flow at low mass ratios. The main objective was to determine whether there is a critical mass ratio for galloping responses, and the corresponding damping ratio effects. To achieve this, a combination of direct simulations and reduced-order models in both its linear and nonlinear versions was used. The work is currently published in the Journal of Fluid Mechanics.

The research team revealed that galloping at a mass ratio was not suppressed but rather delayed to higher flow velocity values. This was illustrated using the linear stability analysis that revealed a close relationship between the galloping-VIV transition and competition between unstable modes. Moreover, galloping could be found at even very low mass ratios, as confirmed by the coupling between the direct simulations and the moving square cylinder. Additionally, for a light structure, an increase in the damping ratio further delays the onset velocity for galloping motion occurrence. Both nonlinear and liner ROMs were found to be useful tools for predicting various qualitative and to some extent quantitative aspects of VIV-galloping phenomenon.

In summary, the study by Dr. Peng Han and Professor Emmanuel de Langre addressed the two fundamental questions about the existence of critical mass ratio for galloping at low mass ratios as well as the role of damping ratios. Contrary to the findings of most previous studies, it was reported that there is no critical mass ratio for galloping of a square cylinder under flow, irrespective of the damping range. In a statement to Advances in Engineering, the authors said that their study would provide useful insights for both structural safety design and energy harvesting from flow.

There is no critical mass ratio for galloping of a square cylinder under flow - Advances in Engineering

About the author

Peng Han completed Ph.D. degree in 2022, jointly at Northwestern Polytechnical University, China and Ecole Polytechnique, Institut Polytechnique de Paris, France (2019-2021). He received his Bachelor’s degree in Mechanical Engineering from the Northwestern Polytechnical University in 2016.

His present research interests are about fluid-structure interactions, particularly flow-induced vibrations, energy harvesting from flow, underwater robots and fluid mechanics.

About the author

Emmanuel de Langre is a full professor in the Department of Mechanics at Ecole Polytechnique, Institut Polytechnique de Paris, France. His research interests are in the broad field of fluid-structure interaction (FSI) and include plant bio-mechanics, energy harvesting based on FSI, vortex-induced vibrations, wind effects and aeroelasticity. He has been since 2014 the Editor-in-chief of the Journal of Fluids and Structures, the leading journal in the field of fluid-structure interactions.

Formerly, he has been a visiting professor at McGill University, Canada, and at Ecole Polytechnique Montreal, Canada. He has been the chair of the Department of Mechanics at Ecole Polytechnique for 9 years. He is also a member of the board of trustees of Ecole Polytechnique.


Han, P., & de Langre, E. (2022). There is no critical mass ratio for galloping of a square cylinder under flowJournal of Fluid Mechanics, 931, A27-20.

Go To Journal of Fluid Mechanics

Han, P., Hémon, P., Pan, G., and de Langre, E. (2021) Nonlinear modeling of combined galloping and vortex-induced vibration of square sections under flow. Nonlinear Dynamics, 103, 3113-3125.

Check Also

Droplet size measurements inside a Venturi - Advances in Engineering

Droplet size measurements inside a Venturi