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AXISYMMETRIC PLUMES IN VISCOUS FLUIDS

Part of: Basic methods in fluid mechanics Hydrodynamic stability

Published online by Cambridge University Press:  17 May 2019

EMMA J. ALLWRIGHT ,
L. K. FORBES Open the ORCID record for L. K. FORBES [Opens in a new window]  and
S. J. WALTERS Open the ORCID record for S. J. WALTERS [Opens in a new window]
EMMA J. ALLWRIGHT
Affiliation:
Mathematics Department, T. U. München, Germany email [email protected] Department of Mathematics and Physics, University of Tasmania, PO Box 37, Hobart, 7001, Tasmania, Australia email [email protected], [email protected]
L. K. FORBES*
Affiliation:
Department of Mathematics and Physics, University of Tasmania, PO Box 37, Hobart, 7001, Tasmania, Australia email [email protected], [email protected]
S. J. WALTERS
Affiliation:
Department of Mathematics and Physics, University of Tasmania, PO Box 37, Hobart, 7001, Tasmania, Australia email [email protected], [email protected]
*
✉[email protected]
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Abstract

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We consider fluid in a channel of finite height. There is a circular hole in the channel bottom, through which fluid of a lower density is injected and rises to form a plume. Viscous boundary layers close to the top and bottom of the channel are assumed to be so thin that the viscous fluid effectively slips along each of these boundaries. The problem is solved using a novel spectral method, in which Hankel transforms are first used to create a steady-state axisymmetric (inviscid) background flow that exactly satisfies the boundary conditions. A viscous correction is then added, so as to satisfy the time-dependent Boussinesq Navier–Stokes equations within the fluid, leaving the boundary conditions intact. Results are presented for the “lazy” plume, in which the fluid rises due only to its own buoyancy, and we study in detail its evolution with time to form an overturning structure. Some results for momentum-driven plumes are also presented, and the effect of the upper wall of the channel on the evolution of the axisymmetric plume is discussed.

Keywords

Boussinesq fluidoverturning plumeunsteady flow

MSC classification

Primary: 76E20: Stability and instability of geophysical and astrophysical flows
Secondary: 76E17: Interfacial stability and instability 76M22: Spectral methods
Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 2 , April 2019 , pp. 119 - 147
Copyright
© 2019 Australian Mathematical Society 

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