Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-24T18:14:19.286Z Has data issue: false hasContentIssue false

A geophysical-scale model of vertical natural convection boundary layers

Published online by Cambridge University Press:  31 July 2008

ANDREW J. WELLS
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK
M. GRAE WORSTER
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

Abstract

A model is developed for turbulent natural convection in boundary layers formed next to isothermal vertical surfaces. A scaling analysis shows that the flow can be described by plume equations for an outer turbulent region coupled to a resolved near-wall laminar flow. On the laboratory scale, the inner layer is dominated by its own buoyancy and the Nusselt number scales as the one-third power of the Rayleigh number (Nu). This gives a constant heat flux, consistent with previous experimental and theoretical studies. On larger geophysical scales the buoyancy is strongest in the outer layer and the laminar layer is driven by the shear imposed on it. The predicted heat transfer correlation then has the Nusselt number proportional to the one-half power of Rayleigh number (Nu) so that a larger heat flux is predicted than might be expected from an extrapolation of laboratory-scale results. The criteria for transitions between flow regimes are consistent with a hierarchy of instabilities of the near-wall laminar flow, with a buoyancy-driven instability operating on the laboratory scale and a shear-driven instability operating on geophysical scales.

Type
Papers
Copyright
Copyright © Cambridge University Press 2008

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

REFERENCES

Ellison, T. H. & Turner, J. S. 1959 Turbulent entrainment in stratified flows. J. Fluid Mech. 6, 423448.CrossRefGoogle Scholar
Gebhart, B., Jaluria, Y., Mahajan, R. L. & Sammakia, B. 1988 Buoyancy Induced Flows and Transport, chap. 11, pp. 547655. Hemisphere.Google Scholar
George, W. K. & Capp, S. P. 1979 A theory for natural convection turbulent boundary layers next to heated vertical surfaces. Intl J. Heat Mass Transfer 22, 813826.CrossRefGoogle Scholar
Hieber, C. A. & Gebhart, B. 1971 Stability of vertical natural convection boundary layers: Expansions at large Prandtl number. J. Fluid Mech. 49, 577591.CrossRefGoogle Scholar
Hölling, M. & Herwig, H. 2005 Asymptotic analysis of the near-wall region of turbulent natural convection flows. J. Fluid Mech. 541, 383397.CrossRefGoogle Scholar
Hughes, G. O. & Griffiths, R. W. 2006 A simple convective model of the global overturning circulation, including effects of entrainment into sinking regions. Ocean Modell. 12, 4679.CrossRefGoogle Scholar
Incropera, F. P. & De Witt, D. P. 2002 Fundamentals of Heat and Mass Transfer, 5th edn. Wiley.Google Scholar
Josberger, E. G. & Martin, S. 1981 A laboratory and theoretical study of the boundary layer adjacent to a vertical melting ice wall in salt water. J. Fluid Mech. 111, 439473.CrossRefGoogle Scholar
Kuiken, H. K. 1968 An asymptotic solution for large Prandtl number free convection. J. Engng Maths 2, 355371.CrossRefGoogle Scholar
Kutateladze, S. S., Kirdyashkin, A. G. & Ivakin, V. P. 1972 Turbulent natural convection on a vertical plate and in a vertical layer. Intl J. Heat Mass Transfer 15, 193202.CrossRefGoogle Scholar
Linden, P. F. 2000 Perspectives in Fluid Mechanics, chapter 6: Convection in the environment, pp. 303321. Cambridge University Press.Google Scholar
McPhee, M. G., Kottmeier, C. & Morison, J. H. 1999 Ocean heat flux in the Central Weddell Sea during winter. J. Phys. Oceanogr. 29, 11661179.2.0.CO;2>CrossRefGoogle Scholar
Morton, B. R., Taylor, G. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 123.Google Scholar
Nachtsheim, P. R. 1963 Stability of free-convection boundary-layer flows. Tech. Rep. NASA TN D–2089.Google Scholar
Ostrach, S. 1952 An analysis of laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force. NACA-TN. 2635.Google Scholar
Papailiou, D. D. 1991 Turbulence models for natural convection flows along a vertical heated plane. Tech. Rep. AGARD-A-R-291 4-1 to 4-5.Google Scholar
Ruckenstein, E. 1998 On the laminar and turbulent free convection heat transfer from a vertical plate over the entire range of Prandtl numbers. Intl Commun. Heat Mass Transfer 25, 10091018.CrossRefGoogle Scholar
Schlichting, H. 1968 Boundary Layer Theory, 6th edn., p. 452 McGraw-Hill.Google Scholar
Taylor, G. I. 1920 Tidal friction in the Irish Sea. Philos. Trans. R. Soc. Lond. A 220, 133.Google Scholar
Tsuji, T. & Nagano, Y. 1988 a Characteristics of a turbulent natural convection boundary layer along a vertical flat plate. Intl J. Heat Mass Transfer 31, 1723–1734. (Data taken from www.ercoftac.org.)CrossRefGoogle Scholar
Tsuji, T. & Nagano, Y. 1988 b Turbulence measurements in a natural convection boundary layer along a vertical flat plate. Intl J. Heat Mass Transfer 31, 2101–2111. (Data taken from www.ercoftac.org.)CrossRefGoogle Scholar
Wells, M. G. & Wettlaufer, J. S. 2005 Two-dimensional density currents in a confined basin. Geophys. Astrophys. Fluid Dyn. 99, 199218.CrossRefGoogle Scholar