Multiple Steady and Oscillatory Solutions in a Collapsible Channel Flow

TL;DR

This study investigates flow in a collapsible channel with elastic walls, revealing multiple steady states and oscillations, and how beam properties influence stability and flow behavior.

Contribution

It demonstrates the coexistence and merging of oscillatory instabilities in a collapsible channel, and analyzes how beam tension and stiffness affect flow stability.

Findings

Multiple steady states can coexist at certain parameters.

Oscillations can occur on both upper and lower steady branches.

Increasing beam tension or stiffness suppresses steady states and oscillations.

Abstract

We study flow driven through a finite-length planar rigid channel by a fixed upstream flux, where a segment of one wall is replaced by a pre-stressed elastic beam subject to uniform external pressure. The steady and unsteady systems are solved using a finite element method. Previous studies have shown that the system can exhibit three steady states for some parameters (termed the upper, intermediate and lower steady branches, respectively). Of these, the intermediate branch is always unstable while the upper and lower steady branches can (independently) become unstable to self-excited oscillations. We show that for some parameter combinations the system is unstable to both upper and lower branch oscillations simultaneously. However, we show that these two instabilities eventually merge together for large enough Reynolds numbers, exhibiting a nonlinear limit cycle which retains characteristics of both the upper and lower branches of oscillations. Furthermore, we show that increasing the beam pre-tension suppresses the region of multiple steady states but preserves the onset of oscillations. Conversely, increasing the beam thickness (a proxy for increasing bending stiffness) suppresses both multiple steady states and the onset of oscillations.