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Stability and energy budget of pressure-driven collapsible channel flows

Published online by Cambridge University Press:  07 September 2011

H. F. Liu ,
X. Y. Luo  and
Z. X. Cai
H. F. Liu
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK
X. Y. Luo*
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK
Z. X. Cai
Affiliation:
Department of Mechanics, Tianjin University, 300072, People’s Republic of China
*
†Email address for correspondence: [email protected]
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Abstract

Although self-excited oscillations in collapsible channel flows have been extensively studied, our understanding of their origins and mechanisms is still far from complete. In the present paper, we focus on the stability and energy budget of collapsible channel flows using a fluid–beam model with the pressure-driven (inlet pressure specified) condition, and highlight its differences to the flow-driven (i.e. inlet flow specified) system. The numerical finite element scheme used is a spine-based arbitrary Lagrangian–Eulerian method, which is shown to satisfy the geometric conservation law exactly. We find that the stability structure for the pressure-driven system is not a cascade as in the flow-driven case, and the mode-2 instability is no longer the primary onset of the self-excited oscillations. Instead, mode-1 instability becomes the dominating unstable mode. The mode-2 neutral curve is found to be completely enclosed by the mode-1 neutral curve in the pressure drop and wall stiffness space; hence no purely mode-2 unstable solutions exist in the parameter space investigated. By analysing the energy budgets at the neutrally stable points, we can confirm that in the high-wall-tension region (on the upper branch of the mode-1 neutral curve), the stability mechanism is the same as proposed by Jensen & Heil. Namely, self-excited oscillations can grow by extracting kinetic energy from the mean flow, with exactly two-thirds of the net kinetic energy flux dissipated by the oscillations and the remainder balanced by increased dissipation in the mean flow. However, this mechanism cannot explain the energy budget for solutions along the lower branch of the mode-1 neutral curve where greater wall deformation occurs. Nor can it explain the energy budget for the mode-2 neutral oscillations, where the unsteady pressure drop is strongly influenced by the severely collapsed wall, with stronger Bernoulli effects and flow separations. It is clear that more work is required to understand the physical mechanisms operating in different regions of the parameter space, and for different boundary conditions.

JFM classification

Instability: Parametric instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 705: Special issue dedicated to Professor T. J. Pedley on his 70th birthday , 25 August 2012 , pp. 348 - 370
Copyright
Copyright © Cambridge University Press 2011

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