Hostname: page-component-5c6d5d7d68-qks25 Total loading time: 0 Render date: 2024-08-15T09:09:25.790Z Has data issue: false hasContentIssue false
  • English
  • Français

Turbulent buoyant convection from a source in a confined two-layered region

Published online by Cambridge University Press:  20 April 2006

Mikio Kumagai
Mikio Kumagai
Affiliation:
Research School of Earth Sciences, The Australian National University, G.P.O. Box 4, Canberra, A.C.T. 2601, Australia Permanent address: Department of Chemical Engineering, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113, Japan.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the time evolution of a layer of fresh water placed on top of a layer of salt water in a laboratory tank when denser salt water is supplied to a nozzle at the free surface. The inflow is carried out slowly so as to form a pure plume, which at first does not penetrate through the interface. The two basic processes that govern the time evolution of the initially fresh-water layer are the filling-box process (Baines & Turner 1969) and the entrainment through the end of the plume which impinges upon the density interface. A theoretical model that takes these two processes into account is presented, with the numerical solution of the asymptotic state, valid at large times. The asymptotic solution and experiment are in good agreement; the theory describes well the vertical buoyancy profile, the change in buoyancy difference across the interface with time, and the time when the plume begins to penetrate through the interface. The entrainment rate obtained from changes in thickness of the upper layer with time can be expressed as a function of the Froude number. The functional dependence is close to Fr3 at small values of Fr, and it approaches a finite limit as Fr increases. The buoyancy flux across the interface, which is non-dimensionalized by the rate of buoyancy input, also changes as a function of Fr, taking a maximum value of 0.168 at Fr = 0.46 and decreasing sharply at larger and smaller Froude numbers. These values agree well with those found from field observations and experiments on the entrainment at the boundary of convectively mixed layers. It is pointed out that some earlier results of Baines (1975) are not consistent with the model presented here.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 147 , October 1984 , pp. 105 - 131
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

Arnold, A. J., Rowland, J. R., Konrad, T. G., Richter, J. H., Jensen, D. R. & Noonkester, V. R. 1975 Simultaneous observations of clear air convection by a pulse radar, an FM—CW radar, an acoustic sounder and an instrumented aircraft. Preprints 16th Radar Meteorology Conf., Houston, pp. 290–295. AMS.
Baines, W. D. 1975 Entrainment by a plume or jet at a density interface. J. Fluid Mech. 68, 309–320.Google Scholar
Baines, W. D. & Turner, J. S. 1969 Turbulent buoyant convection from a source in a confined region. J. Fluid Mech. 37, 51–80.Google Scholar
Ball, F. K. 1960 Control of inversion height by surface heating. Q. J. R. Met. Soc. 86, 483–494.Google Scholar
Caughey, S. J. & Palmer, S. G. 1979 Some aspects of turbulence structure through the depth of the convective boundary layer. Q. J. R. Met. Soc. 105, 811–827.Google Scholar
Deardorff, J. W., Willis, G. E. & Lilly, D. K. 1969 Laboratory investigation of non-steady penetrative convection. J. Fluid Mech. 35, 7–31.Google Scholar
Deardorff, J. W., Willis, G. E. & Stockton, B. H. 1980 Laboratory studies of the entrainment zone of a convectively mixed layer. J. Fluid Mech. 100, 41–64.Google Scholar
Ellison, T. H. & Turner, J. S. 1959 Turbulent entrainment in stratified flow. J. Fluid Mech. 6, 423–448.Google Scholar
Farmer, D. M. 1975 Penetrative convection in the absence of mean shear. Q. J. R. Met. Soc. 101, 869–891.Google Scholar
Germeles, A. E. 1975 Forced plumes and mixing of liquids in tanks. J. Fluid Mech. 71, 601–623.Google Scholar
Hardy, K. R. & Ottersten, H. 1969 Radar investigations of convective patterns in the clear atmosphere. J. Atmos. Sci. 26, 666–672.Google Scholar
Hopfinger, E. J. & Toly, J. A. 1976 Spatially decaying turbulence and its relation to mixing across density interfaces. J. Fluid Mech. 78, 155–175.Google Scholar
Kaimal, J. C., Wyngaard, J. C., Haugen, D. A., Coté, O. R., Izumi, Y., Caughey, S. J. & Readings, C. J. 1976 Turbulence structure in the convective boundary layer. J. Atmos. Sci. 33, 2152–2169.Google Scholar
Kantha, L. H. 1980 Turbulent entrainment at a buoyancy interface due to convective turbulence. In Fjord Oceanography (ed. H. J. Freeland, D. M. Farmer & C. D. Levings), pp. 197–204. Plenum.
Kantha, L. H., Phillips, O. M. & Azad, R. S. 1977 On turbulent entrainment at a stable density interface. J. Fluid Mech. 79, 753–768.Google Scholar
Kato, H. & Phillips, O. M. 1969 On the penetration of a turbulent layer into stratified fluid. J. Fluid Mech. 37, 643–655.Google Scholar
Killworth, P. D. & Carmack, E. C. 1979 A filling-box model of river-dominated lakes. Limnol. Oceanogr. 24, 201–217.Google Scholar
Killworth, P. D. & Turner, J. S. 1982 Plumes with time-varying buoyancy in a confined region. Geophys. Astrophys. Fluid Dyn. 20, 265–291.Google Scholar
Konrad, T. G. 1970 The dynamics of the convective process in clean air as seen by radar. J. Atmos. Sci. 27, 1138–1147.Google Scholar
Linden, P. F. 1973 The interaction of a vortex ring with a sharp density interface: a model for turbulent entrainment. J. Fluid Mech. 60, 467–480.Google Scholar
Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13, 3–23.Google Scholar
Linden, P. F. 1980 Mixing across a density interface produced by grid turbulence. J. Fluid Mech. 100, 691–703.Google Scholar
Lofquist, K. 1960 Flow and stress near an interface between stratified liquid. Phys. Fluids 3, 158–175.Google Scholar
Manins, P. C. 1979 Turbulent buoyant convection from a source in a confined region. J. Fluid Mech. 91, 765–781.Google Scholar
Manins, R. C. & Turner, J. S. 1978 The relation between the flux ratio and energy ratio in convectively mixed layers. Q. J. R. Met. Soc. 104, 39–44.Google Scholar
Moores, W. H., Caughey, S. J., Readings, C. J., Milford, J. R., Mansfield, D. A., Abdula, S., Guymer, T. H. & Johnston, W. B. 1979 Measurements of boundary layer structure and development over SE England using aircraft and tethered balloon instrumentation. Q. J. R. Met. Soc. 105, 397–421.Google Scholar
Morton, B. R. 1956 Buoyant plumes in a moist atmosphere. J. Fluid Mech. 2, 127–144.Google Scholar
Morton, B. R., Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
Sherman, F. S., Imberger, J. & Corcos, G. M. 1978 Turbulence and mixing in stably stratified waters. Ann. Rev. Fluid Mech. 10, 267–288.Google Scholar
Smith, K. A. & Germeles, A. E. 1974 LNG tank stratification consequent to filling procedures. Proc. LNG 4, Alger.
Stull, R. B. 1976a The energetics of entrainment across a density interface. J. Atmos. Sci. 33, 1260–1267.Google Scholar
Stull, R. B. 1976b Mixed layer depth model based on turbulent energetics. J. Atmos. Sci. 33, 1268–1278.Google Scholar
Turner, J. S. 1968 The influence of molecular diffusivity on turbulent entrainment across a density interface. J. Fluid Mech. 33, 639–656.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Turner, J. S. 1981 Small-scale mixing processes. In Evolution of Physical Oceanography (ed. B. A. Warren & C. Wunsch), pp. 236–262. MIT Press.
Warner, J. & Telford, J. W. 1963 Some patterns of convection in the lower atmosphere. J. Atmos. Sci. 20, 313–318.Google Scholar
Warner, J. & Telford, J. W. 1967 Convection below cloud base. J. Atmos. Sci. 24, 374–382.Google Scholar
Willis, G. E. & Deardorff, J. W. 1974 A laboratory model of the unstable planetary boundary layer. J. Atmos. Sci. 31, 1297–1307.Google Scholar
Wolanski, E. J. & Brush, L. M. 1975 Turbulent entrainment across stable density step structures. Tellus 27, 259–268.Google Scholar
Worster, M. G. & Huppert, H. E. 1983 Time-dependent density profiles in a filling box. J. Fluid Mech. 132, 457–466.Google Scholar
Zeman, O. & Tennekes, H. 1977 Parameterization of the turbulent energy budget at the top of the daytime atmospheric boundary layer. J. Atmos. Sci. 34, 111–123.Google Scholar