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Onset of buoyancy-driven convection in Cartesian and cylindrical geometries
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10.1063/1.4801930
/content/aip/journal/pof2/25/4/10.1063/1.4801930
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/4/10.1063/1.4801930

Figures

Image of FIG. 1.
FIG. 1.

(a) Marginal stability (σ = 0) contour in dimensionless wavenumber–time space in an unconfined domain; (b) zoomed-in view around the critical time and critical wavenumber . The base state is unstable in the shaded region above the contour and is stable under it. The dashed line represents the large wavenumber cutoff of . Perturbation modes with wavenumbers larger than this cutoff do not become unstable.

Image of FIG. 2.
FIG. 2.

(a) and (b) vs. or ; (c) and (d) vs. or . The aspect ratio is unity in (a) and (c) so that both and range from 0 to . The dashed line represents or . The behavior of the system is very different depending on whether it lies to the left or the right of the dashed line, as we describe in the text.

Image of FIG. 3.
FIG. 3.

vs. for various values of the aspect ratio : (a) = 10; (b) = 5; (c) = 2; (d) = 0.5; (e) = 0.2; (f) = 0.1. The dashed line represents . Large aspect ratios can significantly reduce boundary effects for both small and large values of .

Image of FIG. 4.
FIG. 4.

vs. in Cartesian geometries at two different aspect ratios: (a) = 1; (b) = 0.1. The dashed line represents . Finer gridding in numerical simulations may be necessary to capture the earliest-growing mode in domains where is small, especially when < 1.

Image of FIG. 5.
FIG. 5.

The indices and of the wavenumber of the earliest-growing mode as a function of the width and the aspect ratio : (a) the index in the direction; (b) the index in the direction. The size in this direction ranges from 0 to ; (c) the index plotted using the same colorbar axis as in (a). The indices generally increase with .

Image of FIG. 6.
FIG. 6.

The indices and of the wavenumber of the earliest-growing mode as a function of the diameter : (a) ; (b) . The earliest-growing mode is usually non-axisymmetric ( ≠ 0).

Image of FIG. 7.
FIG. 7.

Patterns in the plane of the earliest-growing mode for various with = 1: (a) ; (b) ; (c) ; (d) ; (e) ; (f) . Perturbations tend to be highly localized in very positive (red) or very negative (blue) regions. In most cases, there are channels that connect positive regions to each other and other channels that connect negative regions to each other. Pau have observed similar channels in their numerical simulations.

Image of FIG. 8.
FIG. 8.

Patterns in the plane of the earliest-growing mode for at two different aspect ratios: (a) = 2; (b) = 0.5. Patterns are asymmetric with respect to an interchange of and . Two-dimensional modes tend to be preferred when one of the dimensions is small, as it is in (b).

Image of FIG. 9.
FIG. 9.

Top-down and bird's-eye views of patterns in the plane of the earliest-growing mode for smaller values of with : (a), (b) ; (c), (d) ; (e), (f) .

Image of FIG. 10.
FIG. 10.

Top-down and bird's-eye views of patterns in the plane (for clarity, only one quarter of the plane is shown) of the earliest-growing mode for larger values of with : (a), (b) ; (c), (d) ; (e), (f) .

Tables

Generic image for table
Table I.

Zeros (values of ) of the Bessel function derivatives for , = 0, 1, 2, …, 5. Note that . This has implications for Kim , as we discussed in Sec. II B .

Generic image for table
Table II.

The indices and of for various widths with = 1.

Generic image for table
Table III.

The indices , of and the zeros for various diameters .

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/content/aip/journal/pof2/25/4/10.1063/1.4801930
2013-04-25
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Onset of buoyancy-driven convection in Cartesian and cylindrical geometries
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/4/10.1063/1.4801930
10.1063/1.4801930
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