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Revisiting the linear instabilities of plane channel flow between compliant walls

S. Lebbal, F. Alizard and B. Pier

Laboratoire de mécanique des fluides et d'acoustique, École centrale de Lyon – CNRS – Université Claude-Bernard Lyon 1 – INSA, 36 avenue Guy-de-Collongue, 69134 Écully cedex, France

Physical Review Fluids 7, 023903 (32 pages) (2022)

Abstract:

The present investigation revisits the linear stability of Poiseuille channel flow interacting with compliant walls. The results obtained include the dynamics of Tollmien–Schlichting (TS) modes as well as flow-induced surface-instability (FSI) modes, in the form of both traveling-wave flutter (TWF) and divergence (DIV) modes. The compliant wall model con- sists of a spring-backed plate with a viscous substrate deformable in the vertical direction (Davies & Carpenter, J. Fluid Mech. 352, 205–243, 1997). At the interface between the fluid and the walls, the continuity of velocities and stresses, including both viscous and pres- sure contributions, are taken into account. The FSI modes (both varicose and sinuous) and TS modes are then reinterpreted in the light of the two principal nondimensional control parameters: the Reynolds number (Re), which characterizes the base flow, and the reduced velocity (VR), which measures the response of the flexible wall to hydrodynamic loading (De Langre, La Houille Blanche, 3/4, 14–18, 2000). The characteristics of TS and FSI modes are systematically investigated over a large control-parameter space, including wall dissipation, spring stiffness and flexural rigidity. We observe that TWF modes are primarily governed by VR and largely independent of the Reynolds number. It is found that the insta- bility is generally dominated by the TWF mode of varicose symmetry. DIV and TS modes are both affected by VR and Re, confirming that these modes belong to a different class. The onset of the DIV mode is mainly observed for the sinuous motion, when increasing the dissipation. To provide physical insight into the mechanisms driving these instabilities, the perturbative energy equations for both FSI and TS modes are analyzed for a wide range of wall parameters and wavenumbers.

doi:10.1103/PhysRevFluids.7.023903

hal-03337441

2022a_lebbal_prf.pdf

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