Nonlinear Stability Analysis of a Two-Dimensional Model of an Elastic Tube Conveying a Compressible Flow

Abstract

This paper examines the dynamic behavior of a two-dimensional channel whose upper and lower walls deform symmetrically with respect to the center line of the channel. Unsteady fluid dynamic forces acting on the internal wall are analytically evaluated on the basis of a linearized compressible potential flow theory. The effects of distributed springs outside the channel and an internal pressure on the stability characteristics are studied by considering small disturbances about flat and buckled equilibrium configurations of the wall. The analytic methods indicate that no flutter of the flat or buckled wall is predicted when the Mach number is small and the viscous damping coefficient is positive. Numerical results by the Runge-Kutta-Gill method suggest that nonlinear effect of flow should be taken into account to fully examine the dynamic characteristics of the channel conveying a flow.

A numerical simulation of unsteady flow in a two-dimensional collapsible channel

“…An 'eddy doubling' appears at the upper corner at time t = 31.49. This is very similar to the vorticity wave generated in channel flow with a prescribed moving indentation (Pedley & Stephanoff 1985;Ralph & Pedley 1988, 1990. The speed of the vorticity wave crest at position B is approximately 0.348, lower than the mean flow velocity, which is one unit.…”

“…Another important finding is that a train of vorticity waves is generated and propagates downstream during each cycle of oscillation. This is not unexpected, because such waves have previously been experimentally observed downstream of an indentation in a two-dimensional channel which moves in and out in a prescribed way (Pedley & Stephanoff 1985), but this is the first time that they have been found in a coupled flow-structure problem. The phase relation between the waves and the wall motion supports the proposal of Cancelli & Pedley (1985) that the unsteady process of flow separation and the associated energy loss are important parts of the oscillation mechanism.…”

“…Sucking under is also a phenomenon that has been observed in experiments with a thin tube (Ohba et al 1984). It is to avoid the phenomenon that a thick tube is preferred in many experiments on flow in collapsible tubes (Bertram 1982(Bertram ,1986Bertram et al 1990Bertram et al ,1991.…”

Reversal of the Bernoulli Effect and Channel Flutter

“…This type of instability was the only result available to earlier biofluid-mechanical studies based on potential flow models [e.g., Conrad (1969); Matsuzaki & Fung (1979)]. Again, the perturbation considered for divergence is in a symmetric mode.…”

Viscous Flutter of a Finite Elastic Membrane in Poiseuille Flow

“…The tube law was a local function which did not contain any time variable. Matsuzaki & Fung (1977, 1979 considered two-dimensional models in which the #exible wall had a "nite length, but the #ow was again assumed to be a potential #ow. Divergence was invariably predicted, and it was the only instability when structural damping was included.…”