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

Published online by Cambridge University Press:  07 November 2013

Alan Shapiro  and
Evgeni Fedorovich
Alan Shapiro*
Affiliation:
School of Meteorology, University of Oklahoma, Norman, OK 73072, USA Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, OK 73072, USA
Evgeni Fedorovich
Affiliation:
School of Meteorology, University of Oklahoma, Norman, OK 73072, USA
*
†Email address for correspondence: [email protected]
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Abstract

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.

JFM classification

Geophysical and Geological Flows: Baroclinic flows Boundary Layers: Boundary Layers Convection: Convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 736 , 10 December 2013 , pp. 444 - 463
Copyright
©2013 Cambridge University Press 

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