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Modelling gravity currents without an energy closure

Published online by Cambridge University Press:  26 January 2016

N. A. Konopliv ,
Stefan G. Llewellyn Smith ,
J. N. McElwaine  and
E. Meiburg
N. A. Konopliv
Affiliation:
Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
Stefan G. Llewellyn Smith
Affiliation:
Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
J. N. McElwaine
Affiliation:
Department of Earth Sciences, Durham University, Science Laboratories, Durham DH1 3LE, UK
E. Meiburg*
Affiliation:
Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: [email protected]
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Abstract

We extend the vorticity-based modelling approach of Borden & Meiburg (Phys. Fluids, vol. 25 (10), 2013, 101301) to non-Boussinesq gravity currents and derive an analytical expression for the Froude number without the need for an energy closure or any assumptions about the pressure. The Froude-number expression we obtain reduces to the correct form in the Boussinesq limit and agrees closely with simulation data. Via detailed comparisons with simulation results, we furthermore assess the validity of three key assumptions underlying both our as well as earlier models: (i) steady-state flow in the moving reference frame; (ii) inviscid flow; and (iii) horizontal flow sufficiently far in front of and behind the current. The current approach does not require an assumption of zero velocity in the current.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Gravity currents Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 789 , 25 February 2016 , pp. 806 - 829
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Copyright
© 2016 Cambridge University Press 

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References

Anjum, H. J., McElwaine, J. N. & Caulfield, C. P. 2013 The instantaneous Froude number and depth of unsteady gravity currents. J. Hydraul. Res. 51 (4), 432–445.CrossRefGoogle Scholar
Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.CrossRefGoogle Scholar
Birman, V. K., Martin, J. E. & Meiburg, E. 2005 The non-Boussinesq lock-exchange problem. Part 2. High-resolution simulations. J. Fluid Mech. 537, 125–144.Google Scholar
Borden, Z. & Meiburg, E. 2013 Circulation based models for Boussinesq gravity currents. Phys. Fluids 25 (10), 101301.Google Scholar
Chen, C. & Meiburg, E. 2002 Miscible displacements in capillary tubes: influence of Korteweg stresses and divergence effects. Phys. Fluids 14 (7), 2052–2058.Google Scholar
von Kármán, T. 1940 The engineer grapples with nonlinear problems. Bull. Amer. Math. Soc. 46 (8), 615–683.CrossRefGoogle 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
McElwaine, J. N. 2005 Rotational flow in gravity current heads. Phil. Trans. R. Soc. Lond. A 363 (1832), 1603–1623.Google ScholarPubMed
Pozrikidis, C. 1997 Introduction to Theoretical and Computational Fluid Dynamics. Oxford University Press.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Williamson, J. H. 1980 Low-storage Runge–Kutta schemes. J. Comput. Phys. 35 (1), 48–56.Google Scholar