Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T17:04:46.218Z Has data issue: false hasContentIssue false
  • English
  • Français

Sustained gravity currents in a channel

Published online by Cambridge University Press:  10 June 2016

Andrew J. Hogg ,
Mohamad M. Nasr-Azadani ,
Marius Ungarish  and
Eckart Meiburg
Andrew J. Hogg*
Affiliation:
Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK
Mohamad M. Nasr-Azadani
Affiliation:
Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
Marius Ungarish
Affiliation:
Department of Computer Science, Technion, Israel Institute of Technology, Haifa 32000, Israel
Eckart Meiburg
Affiliation:
Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Gravitationally driven motion arising from a sustained constant source of dense fluid in a horizontal channel is investigated theoretically using shallow-layer models and direct numerical simulations of the Navier–Stokes equations, coupled to an advection–diffusion model of the density field. The influxed dense fluid forms a flowing layer underneath the less dense fluid, which initially filled the channel, and in this study its speed of propagation is calculated; the outflux is at the end of the channel. The motion, under the assumption of hydrostatic balance, is modelled using a two-layer shallow-water model to account for the flow of both the dense and the overlying less dense fluids. When the relative density difference between the fluids is small (the Boussinesq regime), the governing shallow-layer equations are solved using analytical techniques. It is demonstrated that a variety of flow-field patterns are feasible, including those with constant height along the length of the current and those where the height varies continuously and discontinuously. The type of solution realised in any scenario is determined by the magnitude of the dimensionless flux issuing from the source and the source Froude number. Two important phenomena may occur: the flow may be choked, whereby the excess velocity due to the density difference is bounded and the height of the current may not exceed a determined maximum value, and it is also possible for the dense fluid to completely displace all of the less dense fluid originally in the channel in an expanding region close to the source. The onset and subsequent evolution of these types of motions are also calculated using analytical techniques. The same range of phenomena occurs for non-Boussinesq flows; in this scenario, the solutions of the model are calculated numerically. The results of direct numerical simulations of the Navier–Stokes equations are also reported for unsteady two-dimensional flows in which there is an inflow of dense fluid at one end of the channel and an outflow at the other end. These simulations reveal the detailed mechanics of the motion and the bulk properties are compared with the predictions of the shallow-layer model to demonstrate good agreement between the two modelling strategies.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Gravity currents Geophysical and Geological Flows: Shallow water flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 798 , 10 July 2016 , pp. 853 - 888
Check for updates
Copyright
© 2016 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

Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.CrossRefGoogle Scholar
Boonkasame, A. & Milewski, P. 2011 The stability of large-amplitude shallow interfacial non-Boussinesq flows. Stud. Appl. Maths 128, 40–58.CrossRefGoogle Scholar
Borden, Z. & Meiburg, E. 2013a Circulation based models for Boussinesq gravity currents. Phys. Fluids 25, 101301.CrossRefGoogle Scholar
Borden, Z. & Meiburg, E. 2013b Circulation-based models for Boussinesq internal bores. J. Fluid Mech. 726, R1.CrossRefGoogle Scholar
Bühler, J., Wright, S. J. & Kim, Y. 1991 Gravity currents advancing into a co-flowing fluid. J. Hydraul. Res. 29, 243–257.Google Scholar
Gratton, J. & Vigo, C. 1994 Self-similar gravity currents with variable inflow revisited: plane currents. J. Fluid Mech. 258, 77–104.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
Hogg, A. J. 2006 Lock-release gravity currents and dam-break flows. J. Fluid Mech. 569, 61–87.CrossRefGoogle Scholar
Hogg, A. J., Hallworth, M. A. & Huppert, H. E. 2005 On gravity currents driven by constant fluxes of saline and particle-laden fluid in the presence of a uniform flow. J. Fluid Mech. 539, 349–385.CrossRefGoogle Scholar
Hogg, A. J. & Woods, A. W. 2001 The transition from inertia to bottom-drag-dominated motion of turbulent gravity currents. J. Fluid Mech. 449, 201–224.CrossRefGoogle Scholar
Hoult, D. P. 1972 Oil spreading on the sea. Annu. Rev. Fluid Mech. 2, 341–368.CrossRefGoogle Scholar
Huppert, H. E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.CrossRefGoogle Scholar
Huppert, H. E. & Simpson, J. E. 1980 The slumping of gravity currents. J. Fluid Mech. 99, 785–799.CrossRefGoogle Scholar
Johnson, C. G. & Hogg, A. J. 2013 Entraining gravity currents. J. Fluid Mech. 731, 477–508.CrossRefGoogle 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
Kranenburg, C. 1993 Gravity-current fronts advancing into horizontal ambient flow. ASCE J. Hydraul. Engng 119, 369–379.CrossRefGoogle Scholar
Long, R. R. 1955 Long waves in a two-fluid system. J. Meteorol. 13, 70–74.2.0.CO;2>CrossRefGoogle Scholar
Meiburg, E., Radhakrishnan, S. & Nasr-Azadani, M. M. 2015 Modeling gravity and turbidity currents: computational approaches and challenges. Appl. Mech. Rev. 67, 04080.CrossRefGoogle Scholar
Nasr-Azadani, M. M., Hall, B. & Meiburg, E. 2013 Polydisperse turbidity currents propagating over complex topography: comparison of experimental and depth-resolved simulation results. Comput. Geosci. 53, 141–153.CrossRefGoogle Scholar
Nasr-Azadani, M. M. & Meiburg, E. 2011 TURBINS: an immersed boundary, Navier–Stokes code for the simulation of gravity and turbidity currents interacting with complex topographies. Comput. Fluids 45, 14–28.CrossRefGoogle Scholar
Ritter, A. 1892 Die Fortpflanzung der Wasserwellen. Z. Verein. Deutsch. Ing. 36 (33), 947–954.Google Scholar
Rottman, J. W. & Simpson, J. E. 1983 Gravity currents produced by instantaneous releases of heavy fluid in a rectangular channel. J. Fluid Mech. 135, 95–110.CrossRefGoogle Scholar
Rotunno, R., Klemp, J. B., Bryan, G. H. & Muraki, D. J. 2011 Models of non-Boussinesq lock-exchange flow. J. Fluid Mech. 675, 1–26.CrossRefGoogle Scholar
Shringapure, M., Lee, H., Ungarish, M. & Balachandar, S. 2013 Front conditions of high-Re gravity currents produced by constant and time-dependent influx: an analytical and numerical study. Eur. J. Mech. (B/Fluids) 41, 109–122.CrossRefGoogle Scholar
Simpson, J. E. 1997 Gravity Currents in the Environment and the Laboratory. p. 244. Cambridge University Press.Google 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. 2009 An Introduction to Gravity Currents and Intrusions. p. 489. CRC Press.CrossRefGoogle Scholar
Ungarish, M. 2011a A non-dissipative solution of Benjamin-type gravity current for a wide range of depth ratios. J. Fluid Mech. 682, 54–65.CrossRefGoogle Scholar
Ungarish, M. 2011b 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. p. 636. Wiley.Google Scholar