The cascade structure of linear instability in collapsible channel flows

Abstract

This paper studies the unsteady behaviour and linear stability of the flow in a collapsible channel using a fluid-beam model. The solid mechanics is analysed in a plane strain configuration, in which the principal stretch is defined with a zero initial strain. Two approaches are employed: unsteady numerical simulations solving the nonlinear fully coupled fluid-structure interaction problem; and the corresponding linearized eigenvalue approach solving the Orr-Sommerfeld equations modified by the beam. The two approaches give good agreement with each other in predicting the frequencies and growth rates of the perturbation modes, close to the neutral curves. For a given Reynolds number in the range of 200-600, a cascade of instabilities is discovered as the wall stiffness (or effective tension) is reduced. Under small perturbation to steady solutions for the same Reynolds number, the system loses stability by passing through a succession of unstable zones, with mode number increasing as the wall stiffness is decreased. It is found that this cascade structure can, in principle, be extended to many modes, depending on the parameters. A puzzling 'tongue' shaped stable zone in the wall stiffness-Re space turns out to be the zone sandwiched by the mode-2 and mode-3 instabilities. Self-excited oscillations dominated by modes 2-4 are found near their corresponding neutral curves. These modes can also interact and form period-doubling oscillations. Extensive comparisons of the results with existing analytical models are made, and a physical explanation for the cascade structure is proposed.

Instabilities in flexible channel flow with large external pressure

“…Simulations for larger N became impractical and were not deemed necessary. This global eigensolver is similar in spirit to that constructed by Luo et al (2008) (see also Liu et al 2009Liu et al , 2012 using a finite element method and an Arnoldi algorithm for computing eigenvalues, but their two-dimensional eigensolver requires extremely large matrices and is impractical for large-scale parameter sweeps. Since the present approach is one-dimensional it has the advantage of much reduced computational cost permitting a unified overview of the parameter space.…”

“…Luo et al 2008;Stewart et al 2009) characterise these modes by the number of turning points in the eigenfunction of the membrane shape,h(x; σ). Hence, according to this convention for T = 2 the mode labelled (i) would be mode-4, the mode labelled (iia) would be mode-5 and the mode labelled (iii) would be mode-6.…”

“…Bertram 2003;Bertram & Tscherry 2006). Models for this system vary in complexity, ranging from spatially one-dimensional models (Stewart et al 2009;Xu et al 2013), rational asymptotic models in the limit of large Reynolds number (Jensen & Heil 2003;Pihler-Puzović & Pedley 2013), eigenfunction expansions based on the local stability and full two-dimensional numerical simulations (Luo & Pedley 1996;Jensen & Heil 2003;Luo et al 2008;Liu et al 2009Liu et al , 2012Xu et al 2014). These models exhibit a variety of oscillatory behaviour depending on how the flow is driven, which could be with either a fixed upstream pressure or a fixed upstream flux.…”

“…Previous models have employed a variety of approaches, including modelling the deformation of the wall using an ad hoc tube law (eg Cancelli & Pedley 1985), as a pre-stressed (ie. tensioned) membrane (eg Luo & Pedley 1996), an elastic beam with finite bending stiffness (eg Luo et al 2008) and a thin elastic shell (eg Jensen & Heil 2003). In this study we model the flexible wall as a tensioned membrane with a large pre-stress, which is assumed to dominate the other elastic restoring forces.…”

“…Conversely, for flow driven by a fixed upstream flux the primary global instability of the system is typically to a so called mode-2 oscillation, where the eigenfunction is inflated at the upstream end of the compliant segment and collapsed at the downstream end (Luo & Pedley 1996;Luo et al 2008). The recent one-dimensional model of Xu et al (2013) (similar to that considered below), with a prescribed external pressure gradient to maintain a uniform basic state, demonstrated that for sufficiently low membrane tension the system is unstable to a family of normal modes; there exists a critical value of the membrane tension above which the uniform state is always stable.…”