Self-excited oscillations of flexible-channel flow with fixed upstream flux - Xu, Feng (2014)
Self-excited oscillations in a collapsible-tube flow driven by fixed upstream flux have been observed by numerical and laboratory experiments. In this thesis we attempt to understand the mechanism of onset of these oscillations by focusing on a reduced physical model. We consider flow in a finite-length planar channel, where a segment of one wall is replaced by a membrane under longitudinal tension. The upstream flux and downstream pressure are prescribed and an external linear pressure distribution is applied to the membrane such that the system admits uniform Poiseuille flow as a steady solution. We describe the system using a one-dimensional model that accounts for viscous and fluid inertial effects. We perform linear stability analysis and weakly nonlinear analysis on the one-dimensional model, the resulting predictions are tested against two-dimensional Navier–Stokes numerical simulation. When the membrane has similar length to the rigid segment of channel downstream of the membrane, we find that in a narrow parameter regime we consider “mode-2” oscillations (i.e. membrane displacements with two extrema) are largely independent of the downstream segment but are driven by divergent instabilities of two non-uniform steady configurations of the membrane. When the downstream segment is much longer than the membrane, our analysis reveals how instability is promoted by a 1:1 resonant interaction between two modes, with the resulting oscillations described by a fourth-order amplitude equation. This predicts the existence of saturated sawtooth oscillations, which we reproduce in full Navier–Stokes simulations of the same system. In this case, our analysis shows some agreements with experimental observations, namely that increasing the length of the downstream tube reduces the frequency of oscillations but has little effect on the conditions for onset. We also use linear stability analysis to show that steady highly-collapsed solutions, constructed by utilizing matched asymptotic expansions, are very unstable, which allows the possibility that they are a precursor to slamming motion whereby the membrane becomes transiently constricted very close to the opposite rigid wall before rapidly recovering.