Transient energy growth in channel flow with compliant walls

Frédéric Alizard (1) , Benoît Pier (1) , Smail Lebbal (1)

1 LMFA - Laboratoire de Mecanique des Fluides et d'Acoustique

In this paper, we present a Lagrangian method for searching initial disturbances which maximise their total energy growth after a certain time horizon for linearized fluid–structure interaction problems. We illustrate this approach for the channel flow case with compliant walls. The walls are represented as thin spring-backed plates, the so-called Kramer-type walls. For nearly critical values of the control parameters (reduced velocity~\VR and Reynolds number~\REY), analyses for sinuous or varicose perturbations show that the fluid–structure system can sustain two types of oscillatory motions of large amplitude. The first motion is associated with two-dimensional perturbations that are invariant in the spanwise direction. For that case and a certain range of streamwise wavenumbers, the short-time dynamics of sinuous perturbations is driven by the nonmodal interaction between the Tollmien–Schlichting (TS) and the travelling-wave flutter (TWF) modes. The amplitude of the oscillation is found to increase with the reduced velocity, and the optimal gain exhibits larger values than its counterpart computed for a channel flow between rigid walls. For perturbations of varicose symmetry, the transient energy is rapidly governed by the unstable TWF mode without a clear low frequency oscillation. The second type of motion concerns streamwise invariant and spanwise periodic perturbations. In that situation, it is found that perturbations of sinuous symmetry exhibit the largest amplification factors. For moderate values of the reduced velocity, VR=O(1), the dynamics is the result of a simple superposition of a standing wave, due to traveling-wave flutter modes propagating downstream and upstream, and the rolls/streaks dynamics. The variations of these oscillations with the reduced velocity, spanwise wavenumber and Reynolds number are then investigated in detail for the sinuous case..

Keywords: Fluid-structure interaction; optimal transient growth; Lagrangian formulation; oscillatory dynamics

Domains: Fluid mechanics [physics.class-ph]

Frédéric Alizard, Benoît Pier, Smail Lebbal. Transient energy growth in channel flow with compliant walls. Physical Review Fluids, 2024, 9, pp.043905. ⟨10.1103/PhysRevFluids.9.043905⟩. ⟨hal-04512520v2⟩