Flow-induced deformations of elastic splitter plates

As a model problem for investigating various flow-induced elastic instabilities, we consider an elastic plate (in grey) clamped on a rigid support (in white) whose shape is varied from a circular cylinder (left figure) to an elliptic bullet (right figure). Such variation in the shape of the rigid support is investigated to obtain various physical mechaniss of flow-induced instabilities: from vortex-induced vibration instabilities when the rigid support is a bluff body (left) to a flutter flag instability when the rigid support is a bluent bodies (right).

The controling parameters of the problems are the Reynolds number, the plate stiffness, the cylinder section, the plate length, and the fluid/solid density ratio. For fixed geometrical parameters and fluid/solid density ratio equal ot 1,  different unstable fluid-solid eigenmodes have been identified in the space of plate stiffness and Reynolds number, as shown in the figure below.

For large values of the rigidity, the unstable eigenmode is the so-called vortex-induced deformation (VIV) mode denoted by the blue region in the figure. For large rigidity, its neutral curve barely depends on the rigidity indicating that this flow-induced elastic instability is merely driven by the Reynolds numbers, the only parameter controlling the fluid dynamics in the problem. More specifically, for the largest values of rigidity explored here, the critical Reynolds number is close to the critical Reynolds number obtained for a rigid splitter plate, which is indicated by the dashed line in the Figure.

This animation of the vortex-induced deformation eigenmode displays a typical hydrodynamic wake pattern downstream the body, that is respnsible for the onset of the vortex-shedding phenomenon behing bluff bodies. Associated to this spatio-temporal structure in the flow, the orange arrows indicate the structural displacement of the splitter plate, that clearly take the form of a structural bending mode. A simple exanimation of the eigenmode can not reveal whether this eigenmode gets unstable as a results of the fluid vortex-shedding exciting the structural bending modes, or the opposite scenario, the bending mode exciting the vortex-shedding. Anyway, such vortex-induced deformation modes get unstable when the structural frequency of the splitter plate (related to the bending stiffiness) gets close the natural vortex-shedding frequency of the wake. This is known as the lock-in effect, since the flow frequency then locks in to the structural frequency. For the low fluid-density ratio investigated here (equal to unity), the lock-in occurs in a large range of structural frequency around the natural frequency of the vortex-shedding.

When decreasing the rigidity of the splitter plate, the vortex-induced deformation mode is stabilized and a new eigenmode gets destabilized for much lower values of the Reynolds number. This new eigenmode, labelled as the strong coupling mode, is unstable in the orange region. Its spatial structure displayed in the left figure is very similar to the one of the vortex-induced deformation, both when examining the structural displacement and the flow pattern. We note however that large magnitude of the fluid compoeent are observed close to the splitter plate.

 

To better understand and distinguish the physical origin of these two eigenmode, whose spatial structures are very similar, it is interesting to examine the corresponding adjoint eigenmode, displayed below.

The adjoint vortex-induced deformation eigenmode, shown on the left figure, displays largest magnitude in the fluid domain, close to the separation points of the underlying base flow on the rigid cylinder. This indicates that the vortex-induced eigenmode is receptive to a fluidic forcing, but not a structural forcing. On the other hand, the adjoint mode associated to the strongly coupled fluid-structure eigenmode, shown in the right figure, exhibits large values both in the fluid and solid domain, close to the tip of te splitter plate. This indicates that this eigenmode is recpetive to either a fluidic or a strucutral forcing, located close the tip of the splitter plate.

The overlapping of the direct and adjoint eigenmodes is a well-known method for determining the wavemaker of hydrodynamic instabilities. Indeed, it determines flow regions where local feedbacks induces largest variation of the eigenvalue. This wavemaker analysis can be extended to fluid-structure eigenmodes. The wavemaker regions of the two unstable eigenmodes described above are displayed in the figure below.

The wavemaker region of the vortex-induced deformation eigenmode is clearly located in the fluid, downtream the splitter plate. On the other hand, the wavemaker region of the coupled fluid-strucutre eigenmode is located in the fluid and solid domains, near the tip of the spitter plate. Such wavemaker analysis is useful to determine the physical origin of fully coupled eigenmodes as it clearly determine physical regions (in the fluid or in the solid) that drive the instability.       

 

Finally, for very small values of the rigidity, a steady eigenmode, labelled a static divergent eigenmode, gets first unstable in the red region. Based on an asymptotic development of the eigenvalue problem around the neutral curve, we have shown that the destabilization of such eigenmode is directly related to a (negative) fluid-induced stiffness that counteracts the (positive) restoring elastic stiffness. When the total stiffness of the system is zero, the instability occurs. More details on that topic can be found in this talk.