Coupled-mode flutter of elongated plates in laminar and turbulent flows

The classical flutter instability  (also known as coupled-mode flutter) arises when two eigenmodes of a structure couple through fluid-structure interaction, leading to high-amplitude unstable motion. Due to its critical implications for the design of airplane wings, it has been extensively studied for over a century [1]. Linear stability has been investigated by several authors, but mainly in the high-Reynolds number range, with simplified flow models (potential flows, piston theory, etc). Using the full Navier-Stokes equations, we revisitate the flutter instability of a 2-DOFs spring-mounted plate at low Reynolds number. We chose a plate of aspect ratio (chord over height) equal to 20. The following analysis is carried out at Re=130 (based on the height), such that the trailing edge recirculation bubble does not exhibit any Von Karmann type instability.

 

Because the translational and rotational springs are mounted at the center of mass of the plate, the structure possesses two classical undamped free-vibration eigenmodes in heaving and pitching motion respectively.

If fluid-structure coupling is suppressed, the spectrum of the coupled system is just the sum of the spectrums of the fluid system and solid system alone. As represented below, it consists in two marginally stable solid eigenmodes (red and black dots) and a forest of stable fluid modes. If the fluid-structure coupling is active, the two solid modes couple to each other through fluid-structure interaction, resulting in a global instability triggered by the red mode.
On a physical point of view, the unstable global mode couples heaving and pitching motions with adequate phasing in order to extract energy from the flow. On the contrary, the black mode dynamics implies that it can only provide energy to the flow.

Contrary to a VIV instability (see above), the flutter mode shows a wake with very large wavelength structures which are convected on very large distances in one period. Those qualitative observation can be linked to the quasi-steady assumption [2], very popular in the FSI community and sometimes used to describe coupled-mode flutter.

            

 

 

By increasing the reduced velocity U* ( = ratio of solid characteristic time over fluid convective time) from 0, one can determine a critical value for which coupled-mode flutter occurs. Close to the flutter boundary, the two modes frequencies tend towards each other, which is the origin of the denomination coupled-mode flutter. However there is no exact coalescence of the frequencies. For higher reduced velocities, the black branch goes zero frequency and gets brutally unstable. This is a static instability often called divergence.

[1] Theodorsen, T., 1935. General Theory of Aerodynamic Instability and the Mechanism of Flutter. NACA Technical Report 496.
[2] Y.C. Fung. 1955 An Introduction to the Theory of Aeroelasticity. New York : Wiley.

 

  • Flow-induced vibrations in turbulent flows

  • Flutter-induced vibration of elongated plates

The classical flutter instability  (also known as coupled-mode flutter) arises when two eigenmodes of a structure couple through fluid-structure interaction, leading to high-amplitude unstable motion. Due to its critical implications for the design of airplane wings, it has been extensively studied for over a century [1]. Linear stability has been investigated by several authors, mainly in the high-Reynolds number range, with simplified flow models (potential flows, piston theory, etc). In a previous study (see above), we investigated the flutter instability in the low Reynolds number regime, by keeping a full Navier-Stokes fluid modelisation. Here, we extend that analysis to turbulent flows by using a RANS framework, closed by the Spalart-Almaras turbulence model.