Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-17T13:19:39.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Models of internal jumps and the fronts of gravity currents: unifying two-layer theories and deriving new results

Published online by Cambridge University Press:  09 May 2018

Marius Ungarish Open the ORCID record for Marius Ungarish [Opens in a new window]  and
Andrew J. Hogg
Marius Ungarish
Affiliation:
Department of Computer Science, Technion, Israel Institute of Technology, Haifa 32000, Israel
Andrew J. Hogg*
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The steady speeds of the front of a gravity current and of an internal jump on a two-layer stratification are often sought in terms of the heights of the relatively dense fluid both up- and downstream from the front or jump, the height of the channel within which they flow, the densities of the two fluids and gravitational acceleration. In this study a unifying framework is presented for calculating the speeds by balancing mass and momentum fluxes across a control volume spanning the front or jump and by ensuring the assumed pressure field is single-valued, which is shown to be equivalent to forming a vorticity balance over the control volume. Previous models have assumed the velocity field is piecewise constant in each layer with a vortex sheet at their interface and invoked explicit or implicit closure assumptions about the dissipative effects to derive the speed. The new formulation yields all of the previously presented expressions and demonstrates that analysing the vorticity balance within the control volume is a useful means of constraining possible closure assumptions, which is arguably more effective than consideration of the flow energetics. However the new approach also reveals that a novel class of models may be developed in which there is shear in the velocity field in the wake downstream of the front or the jump, thus spreading the vorticity over a layer of non-vanishing thickness, rather than concentrating it into a vortex sheet. Mass, momentum and vorticity balances applied over the control volume allow the thickness of the wake and the speed of the front/jump to be evaluated. Results from this vortex-wake model are consistent with published numerical simulations and with data from laboratory experiments, and improve upon predictions from previous formulae. The results may be applied readily to Boussinesq and non-Boussinesq systems and because they arise as simple algebraic expressions, can be straightforwardly incorporated as jump conditions into spatially and temporally varying descriptions of the motion.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Gravity currents Geophysical and Geological Flows: Shallow water flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 846 , 10 July 2018 , pp. 654 - 685
Check for updates
Copyright
© 2018 Cambridge University Press 

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

Baines, P. G. 2016 Internal hydraulic jumps in two-layer systems. J. Fluid Mech. 787, 1–15.Google Scholar
Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.Google Scholar
Borden, Z., Koblitz, T. & Meiburg, E. 2012a Turbulent mixing and wave radiation in non-Boussinesq internal bores. Phys. Fluids 24, 082106.Google Scholar
Borden, Z. & Meiburg, E. 2013a Circulation based models for Boussinesq gravity currents. Phys. Fluids 25, 101301.Google Scholar
Borden, Z. & Meiburg, E. 2013b Circulation-based models for Boussinesq internal bores. J. Fluid Mech. 726, R1.Google Scholar
Borden, Z., Meiburg, E. & Constantinescu, G. 2012b Internal bores: an improved model via the detailed analysis of the energy budget. J. Fluid Mech. 703, 279–314.Google Scholar
Hogg, A. J., Nasr-Azadani, M. M., Ungarish, M. & Meiburg, E. 2016 Sustained gravity currents in a channel. J. Fluid Mech. 798, 853–888.CrossRefGoogle Scholar
Hogg, A. J. & Pritchard, D. 2004 The effects of drag on dam-break and other shallow inertial flows. J. Fluid Mech. 501, 179–212.Google Scholar
Johnson, C. G., Hogg, A. J., Huppert, H. E., Sparks, R. S. J., Phillips, J. C., Slim, A. C. & Woodhouse, M. J. 2015 Modelling intrusions through quiescent and moving ambients. J. Fluid Mech. 771, 370–406.Google Scholar
Von Kármán, T. 1940 The engineer grapples with nonlinear problems. Bull. Am. Math. Soc. 46, 615–683.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.Google Scholar
Klemp, J. B., Rotunno, R. & Skamarock, W. C. 1997 On the propagation of internal bores. J. Fluid Mech. 331, 81–106.CrossRefGoogle Scholar
Konopliv, N. A., Llewellyn-Smith, S. G., McElwaine, J. N. & Meiburg, E. 2016 Modelling gravity currents without an energy closure. J. Fluid Mech. 789, 806–829.Google Scholar
Li, M. & Cummins, F. P. 1998 A note on hydraulic theory of internal bores. Dyn. Atmos. Oceans 28, 1–7.Google Scholar
Simpson, J. E. 1997 Gravity Currents in the Environment and the Laboratory. Cambridge University Press.Google Scholar
Ungarish, M. 2009 An Introduction to Gravity Currents and Intrusions. CRC Press.CrossRefGoogle Scholar
Ungarish, M., Borden, Z. & Meiburg, E. 2014 Gravity currents with tailwaters in Boussinesq and non-Boussinesq systems: two-layer shallow-water dam-break solutions and Navier–Stokes simulations. Environ. Fluid Mech. 14, 451–470.Google Scholar
Wood, I. R. & Simpson, J. E. 1984 Jumps in layered miscible fluids. J. Fluid Mech. 140, 215–231.Google Scholar
Woodhouse, M. J., Phillips, J. C. & Hogg, A. J. 2016 Unsteady turbulent buoyant plumes. J. Fluid Mech. 794, 595–639.Google Scholar