Effect of base-state curvature on self-excited high-frequency oscillations in flow through an elastic-walled channel

Thomas Ward, Robert Whittaker

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We investigate the development of flow-induced small-amplitude high-frequency oscillations occurring in a rigid fluid-conveying channel with a section of the upper wall replaced by a taut elastic sheet. We consider an axially nonuniform base state, caused by a negative transmural pressure (internal minus external), which results in the sheet bending inward, and then examine the evolution of oscillatory perturbations about this state. Due to the curvature of the base state, any normal displacement of the sheet will lead to a change in the axial stretching of the sheet, which is an order of magnitude higher than would be the case for perturbations to a uniform sheet. This stretching provides an additional restoring force that can be dominant. We derive a modified tube law to describe the wall mechanics with this additional effect and combine it with an existing fluid model to obtain a complete description of the system. At leading order, we obtain a one-dimensional eigenvalue problem for the frequencies and mode shapes of the oscillatory perturbations. The nonlinear interaction between the perturbations and the base-state curvature manifests itself as an additional integral term in the eigenvalue problem, corresponding to the total axial stretching in the wall. The normal modes are neutrally stable at leading order, and their slow growth or decay is determined by considering the global energy budget of the system. The stability of the system can be expressed in terms of a critical Reynolds number of the mean flow through the channel. We explain the behavior of the system as two key dimensionless parameters F and K are varied. These quantify the dimensionless axial tension and base-state curvature effects, respectively. Numerical simulations are used to identify distinct flow regimes in parameter space, which we explore further in detail using asymptotic analysis. This reveals that at large K, i.e., strong base-state curvature effects, the leading-order axial profiles of the modes are forced to adjust in order to eliminate the need for significant stretching in the wall. This has the effect of stabilizing both the fundamental and, to a lesser extent, the first harmonic modes, which results in the first harmonic becoming the most unstable mode.
Original languageEnglish
Article number113901
JournalPhysical Review Fluids
Issue number11
Publication statusPublished - 11 Nov 2019

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