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Analytical and numerical results for flow and shock formation in two-layer gravity currents

Published online by Cambridge University Press:  17 February 2009

P. J. Montgomery  and
T. B. Moodie
P. J. Montgomery
Affiliation:
Applied Mathematics Institute, Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada
T. B. Moodie
Affiliation:
Applied Mathematics Institute, Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada
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Abstract

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Many gravity driven flows can be modelled as homogeneous layers of inviscid fluid with a hydrostatic pressure distribution. There are examples throughout oceanography, meteorology, and many engineering applications, yet there are areas which require further investigation. Analytical and numerical results for two-layer shallow-water formulations of time dependent gravity currents travelling in one spatial dimension are presented. Model equations for three physical limits are developed from the hydraulic equations, and numerical solutions are produced using a relaxation scheme for conservation laws developed recently by S. Jin and X. Zin [6]. Hyperbolicity of the model equations is examined in conjunction with the stability Froude number, and shock formation at the interface of the two layers is investigated using the theory of weakly nonlinear hyperbolic waves.

Type
Research Article
Information
The ANZIAM Journal , Volume 40 , Issue 1 , July 1998 , pp. 35 - 58
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Benjamin, T. B., “Gravity currents and related phenomena”, J. Fluid Mech. 31 (1968) 209–248.CrossRefGoogle Scholar
[2]Bonnecaze, R. T., Huppert, H. E. and Lister, J. R., “Particle-driven gravity currents”, J. Fluid Mech. 250 (1993) 339–369.CrossRefGoogle Scholar
[3]D'Alessio, S. J. D., Moodie, T. B., Pascal, J. P. and Swaters, G. E., “Gravity currents produced by sudden release of a fixed volume of heavy fluid”, Stud. Appl. Math. 96 (1996) 359–385.CrossRefGoogle Scholar
[4]D'Alessio, S. J. D., Moodie, T. B., Pascal, J. P. and Swaters, G. E., “Intrusive gravity currents”, Stud. Appl. Math. 98 (1997) 19–46.CrossRefGoogle Scholar
[5]He, Y. and Moodie, T. B., “The signaling problem in nonlinear hyperbolic wave theory”, Stud. Appl. Math. 91 (1991) 215–245.CrossRefGoogle Scholar
[6]Hoff, D., “Invariant regions for systems of conservation laws”, Trans. Amer. Math. Soc. 289 (1985) 591–610.CrossRefGoogle Scholar
[7]Hoult, D. P., “Oil spreading on the sea”, Ann. Rev. Fluid Mech. 4 (1972) 341–368.CrossRefGoogle Scholar
[8]Jin, S. and Xin, Z., “The relaxation schemes for systems of conservation laws in arbitrary space dimensions”, Comm. Pure and Appl. Math. 48 (1995) 235–276.CrossRefGoogle Scholar
[9]von Kármán, T., “The engineer grapples with nonlinear problems”, Bull. Amer. Math. Soc. 46 (1940) 615–683.CrossRefGoogle Scholar
[10]Lawrence, G. A., “On the hydraulics of Boussinesq and non-Boussinesq two-layer flows”, J. Fluid Mech. 215 (1990) 457–480.CrossRefGoogle Scholar
[11]LeVeque, R. J., Numerical Methods for Conservation Laws (Birkhäuser-Verlag, Basel, 1992).CrossRefGoogle Scholar
[12]Majda, A. and Rosales, R., “Resonantly interacting weakly nonlinear hyperbolic waves, I: A single space variable”, Stud. Appl. Math 71 (1984) 149–179.CrossRefGoogle Scholar
[13]Pedlosky, J., Geophysical Fluid Dynamics (Springer-Verlag, New York, 1987).CrossRefGoogle Scholar
[14]Simpson, J. E., Gravity Currents: In the Environment and the Laboratory (Ellis Horwood, Chichester, 1987).Google Scholar
[15]Whitham, G. B., Linear and Nonlinear Waves (J. Wiley and Sons, New York, 1974).Google Scholar