Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-24T20:14:55.225Z Has data issue: false hasContentIssue false
  • English
  • Français

Models of non-Boussinesq lock-exchange flow

Published online by Cambridge University Press:  08 April 2011

R. ROTUNNO ,
J. B. KLEMP ,
G. H. BRYAN  and
D. J. MURAKI
R. ROTUNNO*
Affiliation:
National Center for Atmospheric Research, Boulder, CO 80307, USA
J. B. KLEMP
Affiliation:
National Center for Atmospheric Research, Boulder, CO 80307, USA
G. H. BRYAN
Affiliation:
National Center for Atmospheric Research, Boulder, CO 80307, USA
D. J. MURAKI
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada
*
†Email address for correspondence: rotunno@ucar.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Nearly all analytical models of lock-exchange flow are based on the shallow-water approximation. Since the latter approximation fails at the leading edges of the mutually intruding fluids of lock-exchange flow, solutions to the shallow-water equations can be obtained only through the specification of front conditions. In the present paper, analytic solutions to the shallow-water equations for non-Boussinesq lock-exchange flow are given for front conditions deriving from free-boundary arguments. Analytic solutions are also derived for other proposed front conditions – conditions which appear to the shallow-water system as forced boundary conditions. Both solutions to the shallow-water equations are compared with the numerical solutions of the Navier–Stokes equations and a mixture of successes and failures is recorded. The apparent success of some aspects of the forced solutions of the shallow-water equations, together with the fact that in a real fluid the density interface is a free boundary, shows the need for an improved theory of lock-exchange flow taking into account non-hydrostatic effects for density interfaces intersecting rigid boundaries.

JFM classification

Geophysical and Geological Flows: Gravity currents Geophysical and Geological Flows: Shallow water flows Turbulent Flows: Turbulence simulation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 675 , 25 May 2011 , pp. 1 - 26
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.CrossRefGoogle Scholar
Birman, V., Martin, J. E. & Meiburg, 2005 The non-Boussinesq lock-exchange problem. Part 2. High resolution simulations. J. Fluid Mech. 537, 125–144.CrossRefGoogle Scholar
Bonometti, T., Blachandar, S. & Magnaudet, J. 2008 Wall effects in non-Boussinesq density currents. J. Fluid Mech. 616, 445–475.CrossRefGoogle Scholar
Bryan, G. H. & Rotunno, R. 2008 Gravity currents in a deep anelastic atmosphere. J. Atmos. Sci. 65, 536–556.CrossRefGoogle Scholar
Chorin, A. J. 1967 A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12–26.CrossRefGoogle 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
Courant, R. & Friedrichs, K. O. 1948 Supersonic Flow and Shock Waves. Springer.Google Scholar
Étienne, J., Hopfinger, E. & Saramito, P. 2005 Numerical simulations of high density ratio lock-exchange flows. Phys. Fluids 17, 036601.CrossRefGoogle Scholar
Gröbelbauer, H. P., Fannelop, T. K. & Britter, R. E. 1993 The propagation of intrusion fronts of high density ratios. J. Fluid Mech. 250, 669–687.CrossRefGoogle Scholar
Härtel, C., Meiburg, E. & Necker, F. 2000 Analysis and direct numerical simulation of the flow at a gravity current head. Part 1. Flow topology and front speed for slip and no-slip boundaries. J. Fluid Mech. 418, 189–212.CrossRefGoogle Scholar
John, F. 1981 Partial Differential Equations. Springer.Google Scholar
Keller, J. J. & Chyou, Y.-P. 1991 On the hydraulic lock-exchange problem. J. Appl. Math. Phys. 42, 874–910.Google 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.CrossRefGoogle Scholar
Klemp, J. B., Skamarock, W. C. & Dudhia, J. 2007 Conservative split-explicit time integration methods for the compressible nonhydrostatic equations. Mon. Weath. Rev. 135, 2897–2913.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.CrossRefGoogle Scholar
Moin, P. & Mahesh, K. 1998 Direct numerical simulation: A tool in turbulence research. Annu. Rev. Fluid Mech. 30, 539–578.CrossRefGoogle Scholar
Rottman, J. W. & Simpson, J. E. 1983 Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel. J. Fluid Mech. 135, 95–110.CrossRefGoogle Scholar
Shikhmurazaev, Y. D. 2008 Capillary Flows with Forming Interfaces. Chapman and Hall/CRC.Google Scholar
Shin, J. O., Dalziel, S. & Linden, P. F. 2004 Gravity currents produced by lock exchange. J. Fluid Mech. 521, 1–34.CrossRefGoogle Scholar
Simpson, J. E. & Britter, R. E. 1979 The dynamics of the head of a gravity current advancing over a horizontal surface. J. Fluid Mech. 94, 477–495.CrossRefGoogle Scholar
Ungarish, M. 2007 A shallow-water model for high-reynolds-number gravity currents for a wide range of density differences and fractional depths. J. Fluid Mech. 579, 373–382.CrossRefGoogle Scholar
Ungarish, M. 2008 Energy balances and front speed conditions of two-layer models for gravity currents produced by lock release. Acta Mechanica 201, 63–81.CrossRefGoogle Scholar
Ungarish, M. 2009 An Introduction to Gravity Currents and Intrusions. CRC Press.CrossRefGoogle 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
Ungarish, M. 2011 Two-layer shallow-water dam-break solutions for non-Boussinesq gravity currents in a wide range of fractional depth. J. Fluid Mech. 675, 27–59.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Wicker, L. J. & Skamarock, W. C. 2002 Time-splitting methods for elastic models using forward time schemes. Mon. Weath. Rev. 130, 2088–2097.2.0.CO;2>CrossRefGoogle Scholar
Yih, C.-S. 1965 Dynamics of Nonhomogeneous Fluids. Macmillan.Google Scholar