# Interaction of boundary-layer flow instabilities with flexible elements

We are interested in the interaction between boundary-layer flows and flexible elements. The investigated set-up composes of a two-dimensional (2D) zero-pressure-gradient boundary-layer flow that interacts with a cavity filled by 2D flexible elements (in black) positioned with regular spacing in the streamwise direction. The system is simulated via a finite-elements-method (FEM) that solves the coupled fluid-stuctural problem in the computation domain (gray area).

These *pillars* deform under the boundary-layer flow and their final deformation depends on their gemotry - width and height - and their elasticity properties, in this case the Young modolus E of the material.

The figure shows a detail of the steady-state solution in the upstream part of the cavity. The pillars are clamped to the surface with a spacing equal to their height and their normalised young modulus E/(\rhoU^{2}) is equal to 10, i.e. 16 kPa if we consider a 2 m/s water flow over the flat plate. Hence, even if very flexible, their baseline deformation is still small.

However, the effect of the pillars is not limited only to their deformation. The closed streamlines in the figure highlight a cavity flow inbetween the pillars: this adds to the elastic nature of the setup also 2D porosity. The two effects combined change the stability properties of the flow, as presented in the following.

## Canonical boundary-layer instabilities (TS-waves)

It is largely known that boundary-layers flows become convectively unstable at a critical Re_{x}, i.e. at a critical position in the streamwise direction rescaled according to viscosity and velocity [1]. The instabilities that arise are called Tollmien-Schlichting waves and they present as travelling waves in the boundary layer.

Because of this behaviour, they can be easily identified by computing the optimal inflow condition via resolvant analysis: the animation below shows the optimal response for a angular velocity \omega = 0.06 U/\delta_{0}, where \delta_{0} is the boundary-layer displacement thickness according to the Blasius solution at the upstream end of the cavity.

The convective nature of the instability can be appriciated: the TS wave is excited at the inflow (x=40) and grows while travelling downstream in the flow.

Each frequancy behaves similarly but differently and, since the boundary-layer is a spatially evolving flow, the growth changes with the stream-wise position. This translates in the energy-amplification map that shows how the inflow forcing is amplified at each streamwise position for each forcing frequency.

We can observe two scenarios depending on the excitation frequency: (i) the TS-wave becomes unstable and grows in amplitude all along the domain and (ii) the TS-wave becomes unstable and, after growing in a limited part of the domain, returns stable and reduces its amplitude. The onset of the unstable behaviour is usually called *branch I* and the return to stability *branch II.*

### Interaction of flexible pillars with boundary-layer flow instabilities

Above the canonical asymptotyc stability scenario for a boundary-layer flow has been described: how does it change by the presence of the pillars? The figure below reports the amplification-map when in the presence of the pillars: the cavity extends from x = 0 to x = 20 and hosts 40 flexible pillars with the same density as the fluid (mass ratio M = \rho_{s}/rho_{f} = 1).

The stability properties are modified by the fluid-structure interaction. In a large portion of the parameter space, the TS-wave results amplified with respect to the reference case, but for the higher frequencies - i.e. \omega > 0.10 - the groth results reduced in the cavity region.

The flexibility of the pillars plays a role also in the TS-wave response.

The pillars are moving under the flow perturbation according to their own dynamics, dictated by elasticity (Young modulus E and Poisson coefficient \nu) and inertia (mass-ratio M).

## Other instabilities (work in progress)

The interaction between an elastic wall and a boundary layer can generate other type of instabilities: this is the case of flutter waves (FWs) already observed for compliant surfaces and here reproduced for a clamped elastic wall.

It has been conjectured that this instability rises because of a feedback-loop given by upstream elastic waves in the solid [2].

These elastic-travelling waves are not possible in the case of isolated pillars, since there is no coupling between the pillars but from via fluid. However, when the pillars are clamped on a flexible substrate, collective effects might happen - as represented on the animation below - and FWs might occur.

[1] Schmid, P.J. and Henningson, D.S. (2001) *Stability and transition in shear flows*, Springer

[2] Tsigklifis, K. and Lucey, A.D. (2017) *The interaction of Blasius boundary-layer flow with a compliant panel: global, local and transient analyses*, J. Fluid Mech. 827, pp. 155-193