Hostname: page-component-84b7d79bbc-2l2gl Total loading time: 0 Render date: 2024-07-29T14:22:19.740Z Has data issue: false hasContentIssue false
  • English
  • Français

Gravity current upstream of a buoyant influx in an open-channel flow: a numerical study

Published online by Cambridge University Press:  20 April 2006

Daniel T. Valentine  and
Timothy W. Kao
Daniel T. Valentine
Affiliation:
Mechanical and Industrial Engineering Department, Clarkson College of Technology, Potsdam, New York 13676
Timothy W. Kao
Affiliation:
Civil Engineering Department, The Catholic University of America, Washington, DC 20017
Get access Rights & Permissions [Opens in a new window]

Abstract

The establishment of an upstream intrusion of a buoyant fluid discharged into an open-channel flow of uniform density and finite depth is studied numerically using the full Navier-Stokes and diffusion equations. The problem is posed as an initial-boundary-value problem for the laminar motions of a Boussinesq fluid. The equations are integrated numerically by finite-difference methods. The flow patterns produced are controlled by the influx of buoyancy; therefore they are characterized by an inflow densimetric Froude number. A comparison with available experimental data provides favourable support to the theoretical predictions. The critical value of densimetric Froude number of the source of a vertically downward inflow at the free surface of a channel is determined. For densimetric Froude number less than critical, an intrusion is established on the upstream side of the source. Because dissipative mechanisms associated with viscosity take a finite time to intervene, the intrusion starts as an inviscid gravity current with a propagation speed greater than the surface velocity of the stream. The front speed is proportional to the phase velocity of long internal waves. Subsequently, the front decelerates as the interfacial friction, and, if applicable, the boundary frictional forces increase simultaneously with mass entrainment across the interface. The current slows down towards a two-zone equilibrium: (1) the zone encompassing the current behind the frontal zone, where a steady state is approached with respect to the inertial frame of reference; (2) the frontal zone, where the upstream speed approaches a steady speed of frontal advance, albeit small. The upstream intrusion alters the flow pattern of the ambient stream dramatically. A significant feature of both the upstream and downstream currents is the presence of surface convergence with concomitant downwelling near the fronts. As the upstream front decelerates, wavelike disturbances are excited just behind the front at frequencies characteristic of internal waves. As the front approaches steady state, these disturbances appear to be damped. This problem has practical implications in the design of once-through cooling-water systems for power plants taking their cooling water from rivers.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 140 , March 1984 , pp. 303 - 327
Copyright
© 1984 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

Bata, G. L. 1957 Recirculation of cooling water in rivers and canals. J. Hydraul. Div. ASCE 83 (HY3), 1–27 [Proc. Paper 1265].Google Scholar
Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.Google Scholar
Britter, R. E. & Simpson, J. E. 1978 Experiments on the dynamics of a gravity current head. J. Fluid Mech. 88, 223–240.Google Scholar
Cederwall, K. 1971 Buoyant slot jets into stagnant or flowing environments. W. M. Keck Lab., California Inst. Tech., Rep. KH-R-25.Google Scholar
Didden, N. & Maxworthy, T. 1982 The viscous spreading of plane and axisymmetric gravity currents. J. Fluid Mech. 121, 27–42.Google Scholar
Garvine, R. W. 1974 Dynamics of small-scale oceanic fronts. J. Phys. Oceanogr. 4, 557–569.Google Scholar
Garvine, R. W. 1981 Frontal jump conditions for models of shallow, buoyant surface layer hydrodynamics. Tellus 33, 301–311.Google 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.Google Scholar
Kao, T. W. 1977 Density currents and their applications. J. Hydraul. Div. ASCE 103 (HY5), 543–555 (Proc. Paper 12947].Google Scholar
Kao, T. W., Park, C. & Pao, H.-P. 1978 Inflows, density currents and fronts. Phys. Fluids 21, 1912–1922.Google Scholar
Kármán, T. von 1940 The engineer grapples with non-linear problems. Bull. Am. Math. Soc. 46, 615–683.Google Scholar
Keulegan, G. H. 1958 The motion of saline fronts in still water. Natl Bur. Stand. Rep. 5831.Google Scholar
Lin, Y. J. 1979 A study of inflow characteristics in a stratified fluid. PhD thesis, The Catholic University of America, Washington, D.C.
Lugt, H. & Haussling, H. J. 1974 Laminar flow past an abruptly accelerating elliptic cylinder at 45° incidence. J. Fluid Mech. 65, 711–734.Google Scholar
Pao, H.-P. & Kao, T. W. 1974 Dynamics of establishment of selective withdrawal of a stratified fluid from a line sink. Part 1. Theory. J. Fluid Mech. 65, 657–688.Google Scholar
Polk, E. M., Benedict, B. A. & Parker, F. L. 1971 Cooling water density wedges in streams. J. Hydraul. Div. ASCE 97 (HY10), 1639–1652.Google Scholar
Roache, P. J. 1972 Computational Fluid Dynamics. Hermosa.
Schijf, J. B. & Schönfeld, J. C. 1953 Theoretical considerations on the motion of salt and fresh water. In Proc. Minnesota Intl Hydraul. Convention, University of Minnesota.
Simpson, J. E. 1972 Effects of the lower boundary on the head of a gravity current. J. Fluid Mech. 53 759–768.Google Scholar
Simpson, J. E. 1982 Gravity currents in the laboratory, atmosphere, and ocean. Ann. Rev. Fluid Mech. 14, 213–234.Google Scholar
Stigebrandt, A. 1980 A note on the dynamics of small-scale fronts. Geophys. Astrophys. Fluid Dyn. 16, 225–238.Google Scholar
Torrance, K. E. 1968 Comparison of finite-difference computations of natural convection. J. Res. Natl Bur. Stand. B72, 281–301.Google Scholar
Torrance, K. E. & Rockett, J. A. 1969 Numerical study of natural convection in an enclosure with localized heating from below: creeping flow to the onset of laminar instability. J. Fluid Mech. 36, 33–54.Google Scholar