The Linear Vortex Instability of the Near-vertical Line Source Plume in Porous Media
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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.
KeywordsFree convection Line source Plume Vortex instability
Constant in Eq. 2.16
Acceleration due to gravity
Strength of heat source
- u, v, w
Darcy velocities in the x, y and z-directions
- U, V, W
- x, y, z
Scaled form of η in asymptotic analysis
Coefficient of cubical expansion
Orientation of the plume centreline
Inclination of a horizontal boundary
Orientations of bounding surfaces
Scaled value of x
Other symbols, subscripts and superscripts
Derivative with respect to η
- 0, 1, 2,...
Terms in asymptotic series
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