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Conservation law modelling of entrainment in layered hydrostatic flows

Published online by Cambridge University Press:  05 May 2015

Paul A. Milewski  and
Esteban G. Tabak
Paul A. Milewski*
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
Esteban G. Tabak
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185, USA
*
†Email address for correspondence: p.a.milewski@bath.ac.uk
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Abstract

A methodology is developed for modelling entrainment in two-layer shallow water flows using non-standard conserved quantities, replacing layerwise mass conservation by global energy conservation. Thus, the energy that the standard model would regularly dissipate at internal shocks is instead available to exchange fluid between the layers. Two models are considered for the upper boundary of the flow: a rigid lid and a free surface. The latter provides a selection principle for choosing physically relevant conservation laws among the infinitely many that the former possesses, when the ratio between the baroclinic and barotropic speeds tends to zero. Solutions of the equations are studied analytically and numerically, applied to the lock-exchange problem, and compared with other closures.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Wave breaking
Type
Papers
Information
Journal of Fluid Mechanics , Volume 772 , 10 June 2015 , pp. 272 - 294
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Copyright
© 2015 Cambridge University Press 

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References

Barros, R. 2006 Conservation laws for one-dimensional shallow water models for one and two-layer flows. Math. Models Meth. Appl. Sci. 16, 119–137.Google Scholar
Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.Google Scholar
Boonkasame, A. & Milewski, P. A. 2011 The stability of large-amplitude shallow interfacial non-Boussinesq flows. Stud. Appl. Maths 128, 40–58.CrossRefGoogle Scholar
Camassa, R., Chen, S., Falqui, G., Ortenzi, G. & Pedroni, M. 2012 An inertia ‘paradox’ for incompressible stratified Euler fluids. J. Fluid Mech. 695, 330–340.Google Scholar
Choi, W. & Camassa, R. Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 1–36.Google Scholar
Chumakova, L., Menzaque, F. E., Milewski, P. A., Rosales, R. R., Tabak, E. G. & Turner, C. V. 2009 Stability properties and nonlinear mappings of two and three-layer stratified flows. Stud. Appl. Maths 122, 123–137.CrossRefGoogle Scholar
Esler, J. G. & Pearce, J. D. 2011 Dispersive dam-break and lock-exchange flows in a two-layer fluid. J. Fluid Mech. 667, 555–585.Google Scholar
Holland, D. M., Rosales, R. R., Stefanica, D. & Tabak, E. G. 2002 Internal hydraulic jumps and mixing in two-layer flows. J. Fluid Mech. 470, 63–83.Google Scholar
Huppert, H. E. & Simpson, J. E. 1980 The slumping of gravity currents. J. Fluid Mech. 99, 785–799.CrossRefGoogle Scholar
Jacobsen, T., Milewski, P. A. & Tabak, E. G. 2008 Mixing closures for conservation laws in stratified flows. Stud. Appl. Maths 121, 89–116.CrossRefGoogle Scholar
Klemp, J. B., Rotunno, R. & Skamarock, W. C. 1994 On the dynamics of gravity currents in a channel. J. Fluid Mech. 269, 169–198.Google Scholar
Klemp, J. B., Rotunno, R. & Skamarock, W. C. 1997 On the propagation of internal bores. J. Fluid Mech. 331, 81–106.Google Scholar
Kurganov, A., Noelle, S. & Guergana, P. 2001 Semidiscrete central-upwing schemes for hyperbolic conservation laws and Hamilton–Jacobi equations. SIAM J. Sci. Comput. 23, 707–740.Google Scholar
Li, M. & Cummins, P. F. 1998 A note on hydraulic theory of internal bores. Dyn. Atmos. Oceans 28, 1–7.Google Scholar
Long, R. 1956 Long waves in a two-fluid system. J. Meteorol. 13, 70–74.Google Scholar
Lowe, R. J., Rottman, J. W. & Linden, P. F. 2005 The non-Boussinesq lock-exchange problem. Part 1. Theory and experiments. J. Fluid Mech. 537, 101–124.Google Scholar
Milewski, P. A., Tabak, E. G., Turner, C. V., Rosales, R. R. & Menzaque, F. E. 2004 Nonlinear stability of two-layer flows. Commun. Math. Sci. 2, 427–442.Google Scholar
Rotunno, R., Klemp, J. B., Bryan, G. H. & Muraki, D. J. 2011 Models of non-Boussinesq lock exchange flow. J. Fluid Mech. 675, 1–26.Google Scholar
Shin, J. O., Dalziel, S. B. & Linden, P. F. 2004 Gravity currents produced by lock exchange. J. Fluid Mech. 521, 1–34.Google Scholar
Stoker, J. J. 1958 Water Waves: The Mathematical Theory with Applications. Wiley.Google Scholar
Ungarish, M. 2010 Energy balances for gravity currents with a jump at the interface produced by lock release. Acta Mechanica 211, 1–21.CrossRefGoogle Scholar
Wood, I. R. & Simpson, J. E. 1984 Jumps in layered miscible fluids. J. Fluid Mech. 140, 215–231.Google Scholar