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High-frequency self-excited oscillations in a collapsible-channel flow. (English) Zbl 1049.76015
Summary: High-Reynolds-number asymptotics and numerical simulations are used to describe two-dimensional, unsteady, pressure-driven flow in a finite-length channel, one wall of which contains a section of membrane under longitudinal tension. Asymptotic predictions of stability boundaries for small-amplitude, high-frequency, self-excited oscillations are derived in the limit of large membrane tension. The oscillations are closely related to normal modes of the system, which have a frequency set by a balance between membrane tension and the inertia of the fluid in the entire channel. Oscillations can grow by extracting kinetic energy from the mean Poiseuille flow faster than it is lost to viscous dissipation. Direct numerical simulations, based on a fully coupled finite element discretization of the equations of large-displacement elasticity and the Navier-Stokes equations, support the predicted stability boundaries, and are used to explore larger-amplitude oscillations at lower tensions. These are characterized by vigorous axial sloshing motions superimposed on the mean flow, with transient secondary instabilities being generated both upstream and downstream of the collapsible segment.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76E99 Hydrodynamic stability
76M10 Finite element methods applied to problems in fluid mechanics
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