Steady, supercritical flow in collapsible tubes. Part 2. Theoretical studies

Abstract

Theories are developed to explain the experimental observations of steady, supercritical flow in compliant, partially collapsed tubes, presented in the companion paper (part 1).It is shown that the measured curves of area vs. distance are governed by a combination of (i) friction and gravity, which produce mean gradients of area, and (ii) longitudinal bending and tension forces, which produce standing waves of area superposed upon the mean gradients. The experiments confirm the one-dimensional theory for the mean gradients: (i) in the absence of gravity, friction causes a pressure rise and a positive mean gradient of area; (ii) a downward slope can cancel gravity and lead asymptotically to a uniform state having zero gradients of pressure and area.The inviscid dispersion relationship for area waves due to longitudinal bending and tension is developed, based on a simple, approximate model for the mechanics of the tube. The phase velocity increases as the wavelength decreases, hence the group velocity exceeds the phase velocity. Consequently, in steady flows that are supercritical with respect to the infinite-wavelength phase velocity, energy can propagate upstream and standing waves of area may appear.In the experiments of part 1, longitudinal tension predominated over longitudinal bending. The measured wavelengths of standing waves were found to be in general agreement with the dispersion relationship for tension-induced area waves. The observed streamwise growth of standing area waves is interpreted physically as the attenuation of waves radiating upstream from a source of disturbance such as a shock-like rapid increase of area. The rate of wave attenuation indicates that the skin-friction coefficient has a large out-of-phase oscillatory component. The observed steepness of shock transitions agrees with an approximate theory based on treating the forward portion of the shock as the rearward part of the standing wave train that the shock drives upstream.