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Convectons and secondary snaking in three-dimensional natural doubly diffusive convection
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10.1063/1.4792711
/content/aip/journal/pof2/25/2/10.1063/1.4792711
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/2/10.1063/1.4792711
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Sketch of the vertically extended enclosure. The boundary conditions are no-slip for the velocity everywhere, no-flux for the temperature and concentration on all walls except for = = 0 at = 0 (section represented in light gray) and = = 1 at = 1 (section represented in dark gray).

Image of FIG. 2.
FIG. 2.

Representation of the eigenmodes responsible for the first two instabilities. (a) Marginal eigenmode at the pitchfork bifurcation represented by isosurfaces = ± with chosen appropriately (light indicates = > 0 while dark indicates = − < 0) and by the isovalues of the streamfunction in the plane = 1/2 (light indicates clockwise motion while dark indicates counterclockwise motion). (b1) Similar representation of the marginal eigenmode at the transcritical bifurcation (Fig. 3 ). The last two panels show the evolution of the corresponding nonlinear solution along the branch using isovalues of the streamfunction in the plane = 1/2 at ≈ 810 and ≈ 740. (b2) As in (b1) but showing the branch corresponding to the eigenfunction .

Image of FIG. 3.
FIG. 3.

Bifurcation diagram near the primary transcritical bifurcation showing the kinetic energy as a function of the Rayleigh number along the and branches. The branch bifurcates towards larger but turns around almost immediately in a saddle-node bifurcation that occurs at a very low amplitude. As a result, the two branches appear indistinguishable on the scale of the figure.

Image of FIG. 4.
FIG. 4.

Bifurcation diagrams representing the kinetic energy as a function of the Rayleigh number along the (a) branch and (b) branch. The solutions at each saddle-node are represented in Figs. 5(a) and 5(b) . The branch consists of solutions with an odd number of convection rolls. The oscillations in the branch are associated with the nucleation of new rolls at either side of the structure (left edge of the snaking region) and their growth towards full amplitude (right edge). The repeated nucleation of rolls ends when five rolls are present and the container is full after which the energy increases monotonically with . The branch consists of solutions with an even number of rolls but instead of increasing monotonically after the container is full the branch turns back towards smaller energies and the solution splits into a two-pulse state with a defect in the center of the domain before leaving the snaking region.

Image of FIG. 5.
FIG. 5.

Solutions at successive saddle-nodes in the bifurcation diagrams in Fig. 4 . (a) branch, (b) branch. The solutions are ordered from left to right in terms of increasing distance from the primary bifurcation at ≈ 850.86. The solutions are shown in terms of surfaces of constant vertical velocity, = ±, with chosen appropriately (light indicates = − < 0 while dark indicates = > 0). The last snapshot in (a) is taken at ≈ 841 and that in (b) at ≈ 840. In each roll, the flow is upwards near the = 1 wall and downwards near = 0.

Image of FIG. 6.
FIG. 6.

Representation of the marginal eigenmode generating the branches (a) ( -symmetric), (b) ( -symmetric), (c) ( -symmetric), and (d) ( -symmetric) using a pair of equal and opposite isovalues of the vertical velocity (left panels) and of the component of the vorticity (right panels). To reveal the twisting nature of the eigenmodes, the coordinate axes have been rotated with respect to earlier figures.

Image of FIG. 7.
FIG. 7.

Secondary snaking on the primary branches of localized states in terms of the kinetic energy as a function of the Rayleigh number . (a) , (b) . Secondary bifurcations occur on the subcritical parts of the primary branches and lead to the formation of localized twisted solutions represented by the secondary branches , , , , , and . Snapshots of these solutions are available in Figs. 8 and 9 .

Image of FIG. 8.
FIG. 8.

Snapshots of the secondary branches of solutions shown in Fig. 7 . The snapshots are taken at each saddle-node, beginning at the bifurcation point, and ending at a point on the right of the bifurcation diagram. The same representation as in Fig. 5 is employed. (a) Solutions on the branch ( symmetric, last snapshot at ≈ 837). (b) Solutions on the branch ( symmetric, last snapshot at ≈ 845). (c) Solutions on the branch ( symmetric, last snapshot at ≈ 862). (d) Solutions on the branch ( symmetric, last snapshot at ≈ 864).

Image of FIG. 9.
FIG. 9.

As in Fig. 8 but for (a) and (b). Last snapshots are taken at ≈ 836 and ≈ 832, respectively.

Image of FIG. 10.
FIG. 10.

Slices at = (top) and = − (bottom) of the rightmost states in Figs. 8(c), 8(d), 9(a), and 9(b) , showing isovalues of the vertical velocity , with dark/white indicating positive/negative values. (a) . (b) . (c) . (d) . In each case, is chosen to capture the outermost rolls.

Image of FIG. 11.
FIG. 11.

(a) Time evolution at = 900 of the maximum vertical velocity from a small perturbation of the conduction state proportional to the eigenvector responsible from the transcritical bifurcation at = 850.86. The dashed line shows the corresponding result with imposed symmetry. (b) Solutions at several different times. (c) Solutions at = 150 and = 250 from the direct numerical simulation of Eqs. (2)–(5) with imposed symmetry. All snapshots show isovalues of as in Fig. 5 .

Image of FIG. 12.
FIG. 12.

(a) Time evolution at = 810 of the maximum vertical velocity starting from a solution at ≈ 804 close to the second saddle-node of . The dashed line shows the corresponding result with imposed symmetry. The lower panel shows the evolution of the asymmetry with respect to using the norm , where ≡ (, , , , ) − (, , , , ), measuring the difference between the solution and its image under . When the symmetry is not imposed, the solution decays to the conduction state. This is not the case when imposed. Panels (b) and (c) show snapshots of the solution in these two cases. All snapshots show isovalues of as in Fig. 5 .

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/content/aip/journal/pof2/25/2/10.1063/1.4792711
2013-02-27
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Convectons and secondary snaking in three-dimensional natural doubly diffusive convection
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/2/10.1063/1.4792711
10.1063/1.4792711
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