Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-07-01T22:19:37.974Z Has data issue: false hasContentIssue false
  • English
  • Français

The early stages of shallow flows in an inclined flume

Published online by Cambridge University Press:  25 August 2009

MATTEO ANTUONO ,
ANDREW J. HOGG  and
MAURIZIO BROCCHINI
MATTEO ANTUONO
Affiliation:
DICAT, Università degli Studi di Genova, via Montallegro 1, 16145 Genova, Italy
ANDREW J. HOGG*
Affiliation:
Centre for Environmental & Geophysics Flows, School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
MAURIZIO BROCCHINI
Affiliation:
Istituto di Idraulica e Infrastrutture Viarie, Università Politecnica delle Marche, via Brecce Bianche 12, 60131 Ancona, Italy
*
†Email address for correspondence: A.J.Hogg@Bristol.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The motion of an initially quiescent shallow layer of fluid within an impulsively tilted flume is modelled using the nonlinear shallow water equations. Analytical solutions for the two-dimensional flow are constructed using the method of characteristics and, in regions where neither of the characteristic variables is constant, by adopting hodograph variables and using the Riemann construction for the solution. These solutions reveal that the motion is strongly influenced by the impermeable endwalls of the flume. They show that discontinuous solutions emerge after some period following the initiation of the flow and that for sufficiently long flumes there is a moving interface between wetted and dry regions. Using the hodograph variables we are able to track the evolution of the flow analytically. After the discontinuities develop, we also calculate the velocity and height fields by using jump conditions to express conservation of mass and momentum across the shock and thus we show how the hydraulic jump moves within the domain and how its magnitude grows. In addition to providing the behaviour of the flow in this physical scenario, this unsteady solution also provides an important test case for numerical algorithms designed to integrate the shallow water equations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 633 , 25 August 2009 , pp. 285 - 309
Copyright
Copyright © Cambridge University Press 2009

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

Ancey, C., Iverson, R. M., Rentschler, M. & Denlinger, R. P. 2008 An exact solution for ideal dam-break floods on steep slopes. Water Resour. Res. 44, W01430. doi:10.1029/2007WR006353.CrossRefGoogle Scholar
Armenio, V. & Rocca, M. L. 1996 On the analysis of sloshing of water in rectangular containers: numerical study and experimental validation. Ocean Engng 8, 705–739.CrossRefGoogle Scholar
Carrier, G. F. & Greenspan, H. P. 1958 Water waves of finite amplitude on a sloping beach. J. Fluid Mech. 4, 97–109.CrossRefGoogle Scholar
Faltinsen, O. M., Landrini, M. & Greco, M. 2004 Slamming in marine applications. J. Engng Math 48, 187–217.CrossRefGoogle Scholar
Garabedian, P. R. 1986 Partial Differential Equations. Chelsea Publishing.Google Scholar
Greenspan, H. P. 1958 On the breaking of water waves of finite amplitude on a sloping beach. J. Fluid Mech. 4, 330–334.CrossRefGoogle Scholar
Hogg, A. J. 2006 Lock-release gravity currents and dam-break flows. J. Fluid Mech. 569, 61–87.CrossRefGoogle Scholar
Huang, Z. & Hsiung, C. 1996 Nonlinear shallow-water flow on deck. J. Ship Res. 40, 303–315.CrossRefGoogle Scholar
Kerswell, R. R. 2005 Dam break with Coulomb friction: a model for granular slumping? Phys. Fluids 17, 057101(1–16).CrossRefGoogle Scholar
Lee, T., Zhou, Z. & Cao, Y. 2002 Numerical simulations of hydraulic jumps in water sloshing and water impacting. J. Fluids Engng 124, 215–226.CrossRefGoogle Scholar
Leveque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems, 558 pp. Cambridge.CrossRefGoogle Scholar
Peregrine, D. H. 1972 Equations for water waves and the approximations behind them. In Waves on Beaches and Resulting Sediment Transport (ed. Meyer, R.), chapter 3, pp. 95–121. Academic Press.CrossRefGoogle Scholar
Peregrine, D. H. & Williams, S. M. 2001 Swash overtopping a truncated plane beach. J. Fluid Mech. 440, 391–399.CrossRefGoogle Scholar
Pritchard, D., Guard, P. A. & Baldock, T. E. 2008 An analytical model for bore-driven run-up. J. Fluid Mech. 610, 183–193.CrossRefGoogle Scholar
Shen, M. C. & Meyer, R. E. 1963 Climb of a bore on a beach. Part 3. Run-up. J. Fluid Mech. 16, 113–125.CrossRefGoogle Scholar
Verhagen, J. H. G. & van Wijngaarden, L. 1965 nonlinear oscillations of fluid in a container. J. Fluid Mech. 22, 737–751.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves, 636 pp. Wiley.Google Scholar
Zoppou, C. & Roberts, S. 2003 Explicit schemes for dam-break simulations. J. Hydr. Engng 129, 11–34.CrossRefGoogle Scholar