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The energetics of flow through a rapidly oscillating tube. Part 2. Application to an elliptical tube

Published online by Cambridge University Press:  07 April 2010

ROBERT J. WHITTAKER ,
MATTHIAS HEIL ,
JONATHAN BOYLE ,
OLIVER E. JENSEN  and
SARAH L. WATERS
ROBERT J. WHITTAKER
Affiliation:
Oxford Centre for Industrial and Applied Mathematics, University of Oxford, 24–29 St Giles', Oxford OX1 3LB, UK
MATTHIAS HEIL
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
JONATHAN BOYLE
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
OLIVER E. JENSEN
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
SARAH L. WATERS
Affiliation:
Oxford Centre for Industrial and Applied Mathematics, University of Oxford, 24–29 St Giles', Oxford OX1 3LB, UK
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Abstract

In Part 1 of this work, we derived general asymptotic results for the three-dimensional flow field and energy fluxes for flow within a tube whose walls perform prescribed small-amplitude periodic oscillations of high frequency and large axial wavelength. In the current paper, we illustrate how these results can be applied to the case of flow through a finite-length axially non-uniform tube of elliptical cross-section – a model of flow in a Starling resistor. The results of numerical simulations for three model problems (an axially uniform tube under pressure–flux and pressure–pressure boundary conditions and an axially non-uniform tube with prescribed flux) with prescribed wall motion are compared with the theoretical predictions made in Part 1, each showing excellent agreement. When upstream and downstream pressures are prescribed, we show how the mean flux adjusts slowly under the action of Reynolds stresses using a multiple-scale analysis. We test the asymptotic expressions obtained for the mean energy transfer E from the flow to the wall over a period of the motion. In particular, the critical point at which E = 0 is predicted accurately: this point corresponds to energetically neutral oscillations, the condition which is relevant to the onset of global instability in the Starling resistor.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 648 , 10 April 2010 , pp. 123 - 153
Copyright
Copyright © Cambridge University Press 2010

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References

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