Skip to main content

Similarity models for unsteady free convection flows along a differentially cooled horizontal surface

  • Alan Shapiro (a1) (a2) and Evgeni Fedorovich (a1)

A class of unsteady free convection flows over a differentially cooled horizontal surface is considered. The cooling, specified in terms of an imposed negative buoyancy or buoyancy flux, varies laterally as a step function with a single step change. As thermal boundary layers develop on either side of the step change, an intrinsically unsteady, boundary-layer-like flow arises in the transition zone between them. Self-similarity model solutions of the Boussinesq equations of motion, thermal energy, and mass conservation, within a boundary-layer approximation, are obtained for flows of unstratified fluids driven by a surface buoyancy or buoyancy flux, and flows of stably stratified fluids driven by a surface buoyancy flux. The motion is characterized by a shallow, primarily horizontal flow capped by a weak return flow. Stratification weakens the primary flow and strengthens the return flow. The flows intensify as the step change in surface forcing increases or as the Prandtl number decreases. Simple formulas are obtained for the propagation speeds, trajectories and the evolution of velocity maxima and other local extrema. Similarity-model predictions are verified through numerical simulations in which no boundary-layer approximations are made.

Corresponding author
Email address for correspondence:
Hide All
Amin, N. & Riley, N. 1990 Horizontal free convection. Proc. R. Soc. Lond. A 427, 371384.
Atkinson, B. W. 1981 Meso-Scale Atmospheric Circulations. Academic.
Barenblatt, G. I. 1996 Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press.
Bluman, G. W. & Cole, J. D. 1974 Similarity Methods for Differential Equations. Springer.
Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids, 2nd edn. Oxford University Press.
Chen, T. S., Tien, H. C. & Armaly, B. F. 1986 Natural convection on horizontal, inclined, and vertical plates with variable surface temperature or heat flux. Intl J. Heat Mass Transfer 29, 14651478.
Clifton, J. V. & Chapman, A. J. 1969 Natural-convection on a finite-size horizontal plate. Intl J. Heat Mass Transfer 12, 15731584.
Dayan, A., Kushnir, R. & Ullmann, A. 2002 Laminar free convection underneath a hot horizontal infinite flat strip. Intl J. Heat Mass Transfer 45, 40214031.
Deswita, L., Nazar, R., Ahmad, R., Ishak, A. & Pop, I. 2009 Similarity solutions of free convection boundary layer flow on a horizontal plate with variable wall temperature. Eur. J. Sci. Res. 27, 188198.
Dresner, L. 1983 Similarity Solutions of Nonlinear Partial Differential Equations. Pitman.
Dresner, L. 1999 Applications of Lie’s Theory of Ordinary and Partial Differential Equations. Institute of Physics.
Ede, A. J. 1967 Advances in free convection. In Advances in Heat Transfer, 4 (ed. Hartnett, J. P. & Irvine, T. F. Jr.), pp. 164. Academic.
Fanneløp, T. K. & Webber, D. M. 2003 On buoyant plumes rising from area sources in a calm environment. J. Fluid Mech. 497, 319334.
Fedorovich, E. & Shapiro, A. 2009a Structure of numerically simulated katabatic and anabatic flows along steep slopes. Acta Geophys. 57, 9811010.
Fedorovich, E. & Shapiro, A. 2009b Turbulent natural convection along a vertical plate immersed in a stably stratified fluid. J. Fluid. Mech. 636, 4157.
Fujii, T., Honda, H. & Morioka, I. 1973 A theoretical study of natural convection heat transfer from downward-facing horizontal surfaces with uniform heat flux. Intl J. Heat Mass Transfer 16, 611627.
Garratt, J. R. 1990 The internal boundary layer: a review. Boundary-Layer Meteorol. 50, 171203.
Gebhart, B., Jaluria, Y., Mahajan, R. L. & Sammakia, B. 1988 Buoyancy-Induced Flows and Transport. Hemisphere.
Gill, W. N., Zeh, D. W. & del Casal, E. 1965 Free convection on a horizontal plate. Z. Angew. Math. Phys. 16, 539541.
Higuera, F. J. 1998 Natural convection flow due to a heat source under an infinite horizontal surface. Phys. Fluids 10, 30143016.
Hunt, G. R. & van den Bremer, T. S. 2010 Classical plume theory: 1937–2010 and beyond. IMA J. Appl. Maths 76 (3), 424448.
Ingham, D. B., Merkin, J. H. & Pop, I. 1986 Flow past a suddenly cooled horizontal plate. Wärme-Stoffübertrag. 20, 237241.
Jablonowski, C. & Williamson, D. L. 2011 The pros and cons of diffusion, filters and fixers in atmospheric general circulation models. In Numerical Techniques for Global Atmospheric Models (ed. Lauritzen, P. H., Jablonowski, C., Taylor, M. A. & Nair, R. D.), pp. 389504. Springer.
Kaye, N. B. 2008 Turbulent plumes in stratified environments: a review of recent work. Atmos.-Ocean 46 (4), 433441.
Merkin, J. H. 1985 A note on the similarity solutions for free convection on a vertical plate. J. Engng Maths 19, 189201.
Morton, B. R., Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 123.
Neufeld, J. A., Goldstein, R. E. & Worster, M. G. 2010 On the mechanisms of icicle evolution. J. Fluid Mech. 647, 287308.
Ostrach, S. 1953 An analysis of laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force. NACA Tech. Rep. no. 1111, pp. 63–79.
Pera, L. & Gebhart, B. 1973 Natural convection boundary layer flow over horizontal and slightly inclined surfaces. Intl J. Heat Mass Transfer 16, 11311146.
Rotem, Z. & Claassen, L. 1969 Natural convection above unconfined horizontal surfaces. J. Fluid Mech. 38, 173192.
Samanta, S. & Guha, A. 2012 A similarity theory for natural convection from a horizontal plate for prescribed heat flux or wall temperature. Intl J. Heat Mass Transfer 55, 38573868.
Scase, M. M., Caulfield, C. P., Dalziel, S. P. & Hunt, J. C. R. 2006 Time-dependent plumes and jets with decreasing source strengths. J. Fluid Mech. 563, 443461.
Simpson, J. E. 1987 Gravity Currents: In the Environment and the Laboratory. Ellis Horwood.
Simpson, J. E. 1994 Sea Breeze and Local Wind. Cambridge University Press.
Singh, S. N. & Birkebak, R. C. 1969 Laminar free convection from a horizontal infinite strip facing downwards. Z. Angew. Math. Phys. 20, 454461.
Skamarock, W. C. 2004 Evaluating mesoscale NWP models using kinetic energy spectra. Mon. Weath. Rev. 132, 30193032.
Sparrow, E. M. & Gregg, J. L. 1958 Similar solutions for free convection from a non- isothermal vertical plate. Trans. ASME 80, 379386.
Stewartson, K. 1958 On the free convection from a horizontal plate. Z. Angew. Math. Phys. 9, 276282.
Turner, J. S. 1962 The ‘starting plume’ in neutral surroundings. J. Fluid Mech. 13, 356368.
Woods, A. W. 2010 Turbulent plumes in nature. Annu. Rev. Fluid Mech. 42, 391412.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 16 *
Loading metrics...

Abstract views

Total abstract views: 162 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th March 2018. This data will be updated every 24 hours.