Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-09T22:06:11.421Z Has data issue: false hasContentIssue false
  • English
  • Français

On the front velocity of gravity currents

Published online by Cambridge University Press:  14 August 2007

MARIANO I. CANTERO ,
J. R. LEE ,
S. BALACHANDAR  and
MARCELO H. GARCIA
MARIANO I. CANTERO
Affiliation:
Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
J. R. LEE
Affiliation:
School of Mechanical Engineering, Pusan National University, San 30, Jang Jeon Dong, Kumjung Ku, Busan 609-735, Korea
S. BALACHANDAR*
Affiliation:
Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
MARCELO H. GARCIA
Affiliation:
Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
*
† Author to whom correspondence should be addressed. Present address: Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Highly resolved three-dimensional and two-dimensional simulations of gravity currents in planar and cylindrical configurations are presented. The volume of release of the heavy fluid is varied and the different phases of spreading, namely acceleration, slumping, inertial and viscous phases, are studied. The incompressible Navier–Stokes equations are solved assuming that the Boussinesq approximation is valid for small density difference. The simulations are performed for three different Reynolds numbers (Re): 895, 3450 and 8950 (this particular choice corresponds to values of Grashof number: 105, 1.5 × 106 and 107, respectively). Following their sudden release, the gravity currents are observed to go through an acceleration phase in which the maximum front velocity is reached. As the interface of the current rolls up, the front velocity slightly decreases from the maximum and levels off to a nearly constant value. At higher Re, three-dimensional disturbances grow rapidly and the currents become strongly turbulent. In contrast, in two-dimensional simulations, the rolled-up vortices remain coherent and very strong. Depending on the initial Re of the flow and on the size of the release, the current may transition from the slumping to the inertial phase, or directly to the viscous phase without an inertial phase. New criteria for the critical Re are introduced for the development of the inertial phase. Once the flow transitions to the inertial or viscous phase, it becomes fully three-dimensional. During these phases of spreading, two-dimensional approximations underpredict the front location and velocity. The enhanced vortex coherence of the two-dimensional simulations leads to strong vortex interaction and results in spurious strong time variations of the front velocity. The structure and dynamics of the three-dimensional currents are in good agreement with previously reported numerical and experimental observations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 586 , 10 September 2007 , pp. 1 - 39
Copyright
Copyright © Cambridge University Press 2007

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

Alahyari, A. & Longmire, E. 1996 Development and structure of a gravity current head. Exps Fluids 20, 410–416.CrossRefGoogle Scholar
Allen, J. 1985 Principles of Physical Sedimentology. Allen & Unwin.Google Scholar
Amy, L., Hogg, A., Peakall, J. & Talling, P. 2005 Abrupt transitions in gravity currents. J. Geophys. Res. 110, F03001, doi: 10.1029/2004JF000197.Google Scholar
Balachandar, S. & Parker, S. 2002 Onset of vortex shedding in an inline and staggered array of rectangular cylinders. Phys. Fluids 14 (10), 3714–3732.CrossRefGoogle Scholar
Benjamin, T. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.CrossRefGoogle Scholar
Birman, V., Martin, J. & Meiburg, E. 2005 The non-Boussinesq lock-exchange problem. Part 2. High-resolution simulations. J. Fluid Mech. 537, 125–144.CrossRefGoogle Scholar
Bonnecaze, R. & Lister, J. 1999 Particle-driven gravity currents down planar slopes. J. Fluid Mech. 390, 75–91.CrossRefGoogle Scholar
Bonnecaze, R., Huppert, H. & Lister, J. 1993 Particle-driven gravity currents. J. Fluid Mech. 250, 339–369.CrossRefGoogle Scholar
Bonnecaze, R., Hallworth, M., Huppert, H. & Lister, J. 1995 Axisymmetric particle-driven gravity currents. J. Fluid Mech. 294, 93–121.CrossRefGoogle Scholar
Cantero, M. 2002 Theoretical and numerical modeling of turbidity currents as two-phase flows. MS thesis, University of Illinois at Urbana-Champaign, USA, Urbana, Illinois, USA.Google Scholar
Cantero, M., García, M., Buscaglia, G., Bombardelli, F. & Dari, E. 2003 Multidimensional CFD simulation of a discontinuous density current. In Proc. XXX IAHR Intl Congress, Thessaloniki, Greece, August 24–29, pp. 405–412.Google Scholar
Cantero, M., Balachandar, S., García, M. & Ferry, J. 2006 Direct numerical simulations of planar and cyindrical density currents. J. Appl. Mech. 73, 923–930.CrossRefGoogle Scholar
Canuto, C., Hussaini, M., Quarteroni, A. & Zang, T. 1988 Spectral Methods in Fluid Dynamics. Springer.CrossRefGoogle Scholar
Chakraborty, P., Balachandar, S. & Adrian, R. 2005 On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189–214.CrossRefGoogle Scholar
Choi, S.-U. & García, M. 1995 Modeling of one-dimensional turbidity currents with a dissipative-Galerkin finite element method. J. Hydraul. Res. 33, 623–648.CrossRefGoogle Scholar
Cortese, T. & Balachandar, S. 1995 High performance spectral simulation of turbulent flows in massively parallel machines with distributed memory. Intl J. Supercomputer Applic 9 (3), 187–204.Google Scholar
Daly, B. & Pracht, W. 1968 Numerical study of density-current surges. Phys. Fluids 11 (1), 15–30.CrossRefGoogle Scholar
Deville, M., Fischer, P. & Mund, E 2002 High-Order Methods for Incompressible Fluid Flow. Cambridge University Press.CrossRefGoogle Scholar
Droegemeier, K. & Wilhelmson, R. 1986 Kelvin–Helmholtz instability in a numerically simulated thunderstorm outflow. Bull. Am. Met. Soc. 67, 416–417.Google Scholar
Droegemeier, K. & Wilhelmson, R. 1987 Numerical simulation of thunderstorm outflows dynamics. Part I. Outflow sensitivity experiments and turbulence dynamics. J. Atmos. Sci. 44, 1180–1210.2.0.CO;2>CrossRefGoogle Scholar
Durran, D. 1999 Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer.CrossRefGoogle Scholar
Fannelop, T. & Waldman, G. 1971 The dynamics of oil slicks – or ‘creeping crude’. AIAA J. 41, 1–10.Google Scholar
Fay, J. 1969 The spreads of oil slicks on a calm sea. In Oils in the Sea (ed. David P. Hoult), pp. 53–63.Google Scholar
García, M. 1992 Turbidity currents. In Encyclopedia of Earth System Science, pp. 399–408. Academic.Google Scholar
García, M. & Parker, G. 1989 Experiments on hydraulic jumps in turbidity currents near a canyon-fan transition. Science 245, 393–396.CrossRefGoogle Scholar
García, M. & Parsons, J. 1996 Mixing at the front of gravity currents. Dyn. Atmos. Oceans 24, 197–205.CrossRefGoogle Scholar
Hallworth, M., Huppert, H., Phillips, J. & Sparks, R. 1996 Entrainment into two-dimensional and axisymmetric turbulent gravity currents. J. Fluid Mech. 308, 289–311.CrossRefGoogle Scholar
Hallworth, M., Huppert, H. & Ungarish, M. 2001 Axisymmetric gravity currents in a rotating system: experimental and numerical investigations. J. Fluid Mech. 447, 1–29.CrossRefGoogle Scholar
Härtel, C., Meiburg, E. & Necker, F. 1999 Vorticity dynamics during the start-up phase of gravity currents. Il Nuovo Cimento 22, 823–833.Google Scholar
Härtel, C., Carlsson, F. & Thunblom, M. 2000 a Analysis and direct numerical simulation of the flow at a gravity-current head. Part 2. The lobe-and-cleft instability. J. Fluid Mech. 418, 213–229.CrossRefGoogle Scholar
Härtel, C., Meiburg, E. & Necker, F. 2000 b 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
Hoult, D. 1972 Oil spreading in the sea. Annu. Rev. Fluid Mech. 4, 341–368.CrossRefGoogle Scholar
Huppert, H. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fliud Mech. 121, 43–58.CrossRefGoogle Scholar
Huppert, H. 1998 Quantitative modelling of granular suspensions flows. Phil. Trans. R. Soc. Lond. A 356, 2471–2496.CrossRefGoogle Scholar
Huppert, H. & Simpson, J. 1980 The slumping of gravity currents. J. Fluid Mech. 99, 785–799.CrossRefGoogle Scholar
von Kármán, T. 1940 The engineer grapples with nonlinear problems. Bull. Am. Math. Soc. 46, 615–683.CrossRefGoogle Scholar
Lee, B. & Wilhelmson, R. 1997 a The numerical simulation of non-supercell tornadogenesis. Part I. Initiation and evolution of pretornadic misocyclone circulation along a dry outflow boundary. J. Atmos. Sci. 54, 32–60.2.0.CO;2>CrossRefGoogle Scholar
Lee, B. & Wilhelmson, R. 1997 b The numerical simulation of non-supercell tornadogenesis. Part II. Evolution of a family of tornadoes along a weak outflow boundary. J. Atmos. Sci. pp. 2387–2415.2.0.CO;2>CrossRefGoogle Scholar
Marino, B., Thomas, L. & Linden, P. 2005 The front condition for gravity currents. J. Fluid Mech. 536, 49–78.CrossRefGoogle Scholar
Martin, J. & Moyce, W. 1952 a Part IV. An experimental study of the collapse of liquid columns on a rigid horizontal bottom. Phil. Trans. R. Soc. Lond. A 244, 312–324.Google Scholar
Martin, J. & Moyce, W. 1952 b Part V. An experimental study of the collapse of fluid columns on a rigid horizontal bottom, in a medium of lower, but comparable, density. Phil. Trans. R. Soc. Lond. A 244, 325–334.Google Scholar
Metha, A., Sutherland, B. & Kyba, P. 2002 Interfacial gravity currents. II. Wave excitation. Phys. Fluids 14, 3558–3569.Google Scholar
Mittal, R. & Balachandar, S. 1995 Effect of three-dimensionality on the lift and drag of nominally two-dimensional cylinders. Phys. Fluids 7, 1841–1865.CrossRefGoogle Scholar
Mittal, R. & Balachandar, S. 1996 Direct numerical simulation of flow past elliptic cylinders. J. Comput. Phys. 124, 351–367.CrossRefGoogle Scholar
Necker, F., Härtel, C., Kleiser, L. & Meiburg, E. 2002 High-resolution simulations of particle-driven gravity currents. Intl J. Multiphase Flow 28, 279–300.CrossRefGoogle Scholar
Necker, F., Härtel, C., Kleiser, L. & Meiburg, E. 2005 Mixing and dissipation in particle-driven gravity currents. J. Fluid Mech. 545, 339–372.CrossRefGoogle Scholar
Özgökmen, T. & Chassignet, E. 2002 Dynamics of two-dimensional turbulent bottom gravity currents. J. Phys. Oceanogr. 32, 1460–1478.2.0.CO;2>CrossRefGoogle Scholar
Özgökmen, T., Fischer, P., Duan, J. & Iliescu, T. 2004 Three-dimensional turbulent bottom density currents from a high-order nonhydrostatic spectral element model. J. Phys. Oceanogr. 34, 2006–2026.2.0.CO;2>CrossRefGoogle Scholar
Parsons, J. & García, M. 1998 Similarity of gravity current fronts. Phys. Fluids 10, 3209–3213.CrossRefGoogle Scholar
Patterson, M., Simpson, J., Dalziel, S. & van Heijst, G. 2006 Vortical motion in the head of an axisymmetric gravity current. Phys. Fluids 18 (4), 046601.CrossRefGoogle Scholar
Penney, W., Thornhill, C., Fox, L. & Goodwin, E. 1952 Part III. The dispersion, under gravity, of a column of fluid supported on a rigid horizontal plane. Phil. Trans. R. Soc. Lond. A 244, 285–311.Google Scholar
Rottman, J. & Simpson, J. 1983 Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel. J. Fluid Mech. 135, 95–110.CrossRefGoogle Scholar
Shin, J., Dalziel, S. & Linden, P. 2004 Gravity currents produced by lock exchange. J. Fluid Mech. 521, 1–34.CrossRefGoogle Scholar
Simpson, J. 1972 Effects of the lower boundary on the head of a gravity current. J. Fluid Mech. 53, 759–768.CrossRefGoogle Scholar
Simpson, J. 1982 Gravity currents in the laboratory, atmosphere, and ocean. Annu. Rev. of Fluid Mech. 14, 213–234.CrossRefGoogle Scholar
Simpson, J. 1997 Gravity Currents, 2nd edn. Cambridge University Press.Google Scholar
Simpson, J. & Britter, R. 1979 The dynamics of the head of a gravity current advancing over a horizontal surface. J. Fliud Mech. 94, 477–495.CrossRefGoogle Scholar
Spicer, T. & Havens, J. 1987 Gravity flow and entrainment by dense gases released instantaneously into calm air. In Proc. Third Intl Symp. on Stratified Flows Pasadena, California, USA, February 3–5 (ed. E. List & G. Jirka), pp. 642–651.Google Scholar
Terez, D. & Knio, O. 1998 a Numerical simulation of large-amplitude internal solitary waves. J. Fluid Mech. 362, 53–82.CrossRefGoogle Scholar
Terez, D. & Knio, O. 1998 b Numerical study of the collapse of an axisymmetric mixed region in a pycnoclyne. Phys. Fluids 10, 1438–1448.CrossRefGoogle Scholar
Thomas, L., Dalziel, S. & Marino, B. 2003 The structure of the head of an inertial gravity current determined by particle-tracking velocimetry. Exps Fluids 34, 708–716.CrossRefGoogle Scholar
Ungarish, M. & Zemach, T. 2003 On axisymmetric rotating gravity currents: two-layer shallow-water and numerical solutions. J. Fluid Mech. 481, 37–66.CrossRefGoogle Scholar
Zhou, J., Adrian, R., Balachandar, S. & Kendall, T. 1999 Mechanics for generating coherent packets of hairpin vortices. J. Fluid Mech. 387, 353–396.CrossRefGoogle Scholar