Reversal of the Bernoulli effect and channel flutter

Abstract

This paper is concerned with the collapse and subsequent self-excited oscillation of compliant tubes conveying fluids. Our model considers a two-dimensional, in viscid, shear flow in a flexible channel of infinite length subject to linear, travelling varicose waves. Analysis of the boundary-value problem leads to two findings which do not seem to have been noticed before, despite the close attention this kind of fluid-structure interaction has attracted on account of its medical significance. The pressure perturbation on the wall has two components, the first is in-phase with the wall displacement and the second with the velocity of the wall motion. For potential flow, the first component is the only one tending to destabilize and is known as the Bernoulli effect. For shear flow, however, the sign of the pressure is reversed as the Bernoulli effect is overcome by the perturbations of the vorticity field. Streamline patterns show that Kelvin's "cats' eyes" are sheltered in the wider channel sections, rendering the effective flow passage smaller where the physical width is larger. The second component produces a wave drag, hence irreversible transfer of energy from the flow to waves. We argue that this is a possible mechanism for the self-excited oscillation observed in experiments. This mechanism is similar to Miles's (1957) mechanism of water wave generation by wind, which is a class B instability according to the Benjamin-Landahl categorization, but the accompanying reversal of the Bernoulli effect is different and depends essentially on the presence of a second boundary. The eigenvalue problem is also considered and it is shown that dynamic instability of long but finite wavelength could be experienced by compliant channels with thick walls, a typical application being the respiratory flow in the upper airways. The critical flow speed is given in terms of the channel properties. © 1998 Academic Press Limited.