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IMA Journal of Applied Mathematics Advance Access published online on November 13, 2007
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxm050
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

The development of slugging in two-layer hydraulic flows

D. J. Needham

School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK

J. Billingham{dagger}

School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK

R. M. S. M. Schulkes

Norsk Hydro, Oil and Energy Research Centre, Porsgrunn N3908, Norway

A. C. King

School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK

{dagger} Email: john.billingham{at}nottingham.ac.uk

Received on March 1, 2007; Revision received May 11, 2007. Accepted on August 16, 2007

In this paper, we develop a hydraulic theory to describe the occurrence and structure of slugging in a confined two-layer gas–liquid flow generated by prescribed, constant, upstream volumetric flow rates in each layer. A linearized theory for the uniform flow is established, after which we use bifurcation theory to study fully non-linear periodic travelling wave structures. We find that a two-parameter family of such travelling wave solutions exists. Under given conditions, the volumetric flow rate constraint provides a relation between these two parameters. To select a unique periodic travelling wave solution, we require a further relation. We first investigate the conjecture that the periodic travelling wave solution selected in the initial value problem has the same wavelength as the linearly most temporally unstable mode. To do this, we solve the initial value problem numerically on a periodic domain. We find that the separation of the liquid slugs that form is much longer than the wavelength of the most unstable temporal mode. We then develop a different conjecture based on the convective instability of the long ‘tails’ of the available periodic travelling wave solutions, which leads to a better understanding of the wavelength selection process.

Keywords: Multiphase flow; stratified pipeline flow; roll waves.


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