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The Linear Vortex Instability of the Near-vertical Line Source Plume in Porous Media

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Abstract

Free convection plumes usually rise vertically, but do not do so when in an asymmetrical environment. In such cases they are susceptible to a thermoconvective instability because warmer fluid lies below cooler fluid in the upper half of the plume. We analyse the behaviour of streamwise vortex disturbances in plumes that are close to being vertical. The linearised equations subject to the boundary layer approximation are parabolic and are solved using a marching method. Our computations indicate that disturbances tend to be centred in the upper half of the plume. A neutral curve is determined and an asymptotic theory is developed to describe the right hand branch of this curve. The left hand branch is not amenable to an asymptotic analysis, and it is found that the onset of convection for small wavenumbers is very sensitively dependent on both the profile of the initiating disturbance and where it is introduced.

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Abbreviations

c :

Constant in Eq. 2.16

C p :

Specific heat

d :

Natural lengthscale

f :

Reduced streamfunction

g :

Acceleration due to gravity

k :

Vortex wavenumber

K :

Permeability

p :

Fluid pressure

P :

Perturbation pressure

q′′′:

Strength of heat source

T :

Temperature

u, v, w :

Darcy velocities in the x, y and z-directions

U, V, W :

Perturbation velocities

x, y, z :

Cartesian coordinates

Y :

Scaled form of η in asymptotic analysis

α :

Thermal diffusivity

β :

Coefficient of cubical expansion

δ :

Orientation of the plume centreline

ϵ:

Inclination of a horizontal boundary

μ :

Fluid viscosity

θ :

Non-dimensional temperature

Θ:

Perturbation temperature

\(\phi\) :

Angular coordinate

\(\phi^+,\phi^-\) :

Orientations of bounding surfaces

ψ :

Streamfunction

ρ :

Fluid density

η :

Similarity variable

ξ :

Scaled value of x

−:

Dimensional

^:

Non-dimensional

′:

Derivative with respect to η

c :

Critical value

∞:

Ambient

0, 1, 2,...:

Terms in asymptotic series

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Authors and Affiliations

  1. Department of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    D. Andrew S. Rees

  2. Department of Mechanical Engineering, University of Bath, Bath, BA2 7AY, UK

    D. Andrew S. Rees

  3. Department of Thermal Engineering and Fluid Mechanics, Transilvania University, Bd. Eroilor 29, Brasov, 500036, Romania

    Adrian Postelnicu

  4. School of Mathematics and Statistics, University of Western Australia, Crawley, WA, 6009, Australia

    Andrew P. Bassom

Authors
  1. D. Andrew S. Rees
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  2. Adrian Postelnicu
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  3. Andrew P. Bassom
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Correspondence to D. Andrew S. Rees.

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Rees, D.A.S., Postelnicu, A. & Bassom, A.P. The Linear Vortex Instability of the Near-vertical Line Source Plume in Porous Media. Transp Porous Med 74, 221–238 (2008). https://doi.org/10.1007/s11242-007-9195-3

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  • DOI: https://doi.org/10.1007/s11242-007-9195-3

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