Abstract
We model the unsteady evolution of turbulent buoyant plumes following temporal changes to the source conditions. The integral model is derived from radial integration of the governing equations expressing the evolution of mass, axial momentum and buoyancy in the plume. The nonuniform radial profiles of the axial velocity and density deficit in the plume are explicitly captured by shape factors in the integral equations; the commonlyassumed tophat profiles lead to shape factors equal to unity. The resultant model for unsteady plumes is hyperbolic when the momentum shape factor, determined from the radial profile of the mean axial velocity in the plume, differs from unity. The solutions of the model when source conditions are maintained at constant values are shown to retain the form of the wellestablished steady plume solutions. We demonstrate through a linear stability analysis of these steady solutions that the inclusion of a momentum shape factor in the governing equations that differs from unity leads to a wellposed integral model. Therefore, our model does not exhibit the mathematical pathologies that appear in previously proposed unsteady integral models of turbulent plumes. A stability threshold for the value of the shape factor is also identified, resulting in a range of its values where the amplitude of small perturbations to the steady solutions decay with distance from the source. The hyperbolic character of the system of equations allows the formation of discontinuities in the fields describing the plume properties during the unsteady evolution, and we compute numerical solutions to illustrate the transient development of a plume following an abrupt change in the source conditions. The adjustment of the plume to the new source conditions occurs through the propagation of a pulse of fluid through the plume. The dynamics of this pulse are described by a similarity solution and, through the construction of this new similarity solution, we identify three regimes in which the evolution of the transient pulse following adjustment of the source qualitatively differ.
Original language  English 

Pages (fromto)  595638 
Number of pages  44 
Journal  Journal of Fluid Mechanics 
Volume  794 
Early online date  5 Apr 2016 
DOIs  
Publication status  Published  May 2016 
Keywords
 Plumes/thermals
 Turbulence modelling
 Turbulent convection
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Profiles

Professor Andrew J Hogg
 School of Mathematics  Professor of Fluid Mechanics
 Cabot Institute for the Environment
 Fluids and materials
 Applied Mathematics
Person: Academic , Member

Dr Jeremy C Phillips
 School of Earth Sciences  Reader in Physical Volcanology
 Cabot Institute for the Environment
 Volcanology
Person: Academic , Member

Dr M J Woodhouse
 School of Earth Sciences  NERC Knowledge Exchange Fellow
 Cabot Institute for the Environment
 Applied Mathematics
Person: Academic , Member