الملخص الإنجليزي
The dynamics of a column of buoyant fluid rising in a less buoyant fluid bounded by two fixed vertical walls a finite distance apart is investigated, as an example of a compositional plume in a bounded domain. This is an extension to a previously studied plume in the absence of the sidewalls. The whole system is rotating uniformly about the vertical. The problem is governed by six dimensionless parameters: the Grashof number, R , which is the ratio of the buoyancy force due to the difference in concentration of light material of the plume and that of the surrounding fluid to the viscous force, the rotation parameter, t, which is a measure of the Coriolis force relative to the viscous force, the Prandtl number, o, which is the ratio of viscosity, v , to thermal diffusivity, K, the thickness of the plume, 2 X,, the distance, d , between the two vertical walls and the distance, , between the plume and the nearest sidewall made dimensionless using the salt-finger length scale as a unit of length. The main objective of this study is to investigate the influence of the boundary on the solution obtained when the surrounding fluid is unbounded. The mean (basic) state solution is independent of R, T and o. The symmetry of the solution present in the absence of the boundaries is here lost unless the plume lies half-way between the two sidewalls. The fluxes of material, heat and buoyancy are here strongly modified by the new parameters d and ag. The material and buoyancy fluxes possess local maxima in the (xo, az - xo) plane for moderate to small thicknesses of the plume when the plume lies close to the wall. This has the consequence of changing the stability properties obtained in the absence of the boundaries. In the absence of the boundaries, the plume is always unstable when R is small and the growth rate is O(R). The instability takes the form of one of two uncoupled categories of solutions: one of which is even in x, referred to as the varicose (V) mode, and the other odd in x and is referred to as the sinuous (S) mode. In the absence of rotation, the introduction of the boundaries introduces dramatic changes to the stability of the plume. A region of instability with a growth rate of 0(1) appears when the plume is thin and lies close to the : boundary. Moreover, a region of stability appears.when the plume is close to the boundary but has a large thickness. For other regions of the plane (xo, az – X), the plume has a growth rate of the same order of magnitude as in the absence of the boundaries but its magnitude is 'reduced as the distance, d, between the sidewalls is decreased. Instability again takes the form of one of two categories of solutions which related to the S and V modes but here modified by the presence of the boundaries and the position of the plume relative to the nearest boundary. In addition, the presence of the boundary suppresses the three-dimensional instabilities present in the unbounded domain and allows only two-dimensional instabilities for moderate to small distances between the bounding walls.
When rotation is present, the influence of the boundaries on the Cartesian plume is less dramatic. Here the growth rate of O(1) found in the absence of rotation, persists but the magnitude of the growth rate increases as the plume approaches a sidewall. Another influence of the presence of the boundaries in the presence of rotation is that the modified varicose (MV) mode is suppressed and only the modified sinuous (MS) mode is unstable.
Keywords: compositional plumes, stability, growth rate, bounded domain, rotation.
