Revisiting the linear instabilities of plane channel flow between compliant walls
Résumé
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 modes. The compliant wall model consists of a spring-backed plate with a viscous substrate deformable in the vertical direction [C. Davies and P. W. Carpenter, J. Fluid Mech. 352, 205 (1997)]. At the interface between the fluid and the walls, the continuity of velocities and stresses, including both viscous and pressure 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 (V R), which measures the response of the flexible wall to hydrodynamic loading [E. De Langre, La Houille Blanche, 3, 14 (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 V R and largely independent of the Reynolds number. It is found that the instability is generally dominated by the TWF mode of varicose symmetry. Divergence and TS modes are both affected by
V R and Re, confirming that these modes belong to a different class. The onset of the divergence 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
Origine : Fichiers éditeurs autorisés sur une archive ouverte
