Experimental results are presented for steady, supercritical flow of a liquid in a thin-walled compliant tube which is in a state of partial collapse due to a negative transmural pressure. Particular attention is paid to the effects of longitudinal tension.With a constant external pressure, friction acts to increase the area in the down-stream direction. With the tube tilted downward, friction may be so balanced by gravity forces as to result in an asymptotic approach to an equilibrium situation in which the area and velocity remain constant. When the downstream pressure is increased sufficiently, shock-like transitions to subcritical inflated states, with positive transmural pressure, occur. The longitudinal length scale of the shock is on the order of one to several tube diameters. The pressure rise across the shock lies between that for loss-free pressure recovery and that given by the Borda-Carnot sudden-expansion theory.The presence of longitudinal tension causes a train of standing waves of area to appear upstream of a local area disturbance, such as a shock-like transition. The standing waves are superimposed upon the more gradual area changes associated with friction and gravity. The wave amplitude grows in the downstream direction. Theoretical interpretations of the observations are presented in the companion paper (part 2).
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.
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