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Numerical simulations of nonlinear thermally stratified spin-up in a circular cylinder
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10.1063/1.3505025
/content/aip/journal/pof2/22/11/10.1063/1.3505025
http://aip.metastore.ingenta.com/content/aip/journal/pof2/22/11/10.1063/1.3505025

Figures

Image of FIG. 1.
FIG. 1.

Schematic of the fluid system. At , the cylinder is instantly accelerated from the initial rotation rate to a new rotation rate . The inset shows the formation of the corner region during the first stage of spin-up at .

Image of FIG. 2.
FIG. 2.

Comparison of stratified spin-up experiments with numerical simulations for the azimuthal velocity . (a) Salinity stratification: the fluid depth is , tank radius is , and the buoyancy frequency is . The spin-up sequence is at . The vertical position is . The symbols are the results of laboratory measurements in Ref. 21 and the dashed lines (− −) are our numerical simulations. (b) Thermal stratification—parameters: , , , , , , , and . The vertical and radial locations are at the mid-depth and . The circles (○) correspond to the laser-Doppler measurements in Ref. 31 and the solid lines (—) to our numerical simulations.

Image of FIG. 3.
FIG. 3.

Isotherms on the planes . At , there are 32 linearly spaced contour levels in the range . (a) LBU: the region shown is and (not to scale). (b) SRO: the region shown is and (not to scale). Movies 3, available in the online version, show the spatiotemporal characteristics for the isotherms on the planes over several rotation periods at a rate of 10 frames/s, with each frame being one rotation period apart (enhanced online). [URL: http://dx.doi.org/10.1063/1.3505025.1] [URL: http://dx.doi.org/10.1063/1.3505025.2]10.1063/1.3505025.110.1063/1.3505025.2

Image of FIG. 4.
FIG. 4.

LBU: axial vorticity ; there are seven positive (solid/black) and seven negative (white/dashed) linearly spaced contour levels in the range . The region shown is and .

Image of FIG. 5.
FIG. 5.

SRO: axial vorticity . There are seven positive (solid/black) and seven negative (white/dashed) linearly spaced contour levels in the range . The region shown is and . Movies 5, available in the online version, show the spatiotemporal characteristics for the vertical vorticity (enhanced online). [URL: http://dx.doi.org/10.1063/1.3505025.3] [URL: http://dx.doi.org/10.1063/1.3505025.4] [URL: http://dx.doi.org/10.1063/1.3505025.5]10.1063/1.3505025.310.1063/1.3505025.410.1063/1.3505025.5

Image of FIG. 6.
FIG. 6.

NEU: isotherms on the planes ; at , there are 15 linearly spaced contour levels in the range showing the formation of the corner regions and the motion after the vortex core is formed. The region shown is and . Movie 6, available in the online version, shows the spatiotemporal characteristics for the isotherms on the planes over 145 rotation periods at a rate of 10 frames/s, with each frame being one rotation period apart (enhanced online). [URL: http://dx.doi.org/10.1063/1.3505025.6]10.1063/1.3505025.6

Image of FIG. 7.
FIG. 7.

NEU: vertical profiles of the radial velocity and temperature at different radial locations and . (a) Radial velocity at times (○), 3 (◻), 7 (▽), and 20 (△). Note that the solid line indicates the depth of the Ekman layer. [(b)–(d)] Temperature profiles at (− −), 3 (○), 20 (△), 45 (◇), and 120 (×).

Image of FIG. 8.
FIG. 8.

NEU: temperature and vortex structure identified by the isosurfaces of colored by temperature. The figures on the left correspond to and on the middle/right to the vortex structure. The region shown is and . Movie 8, available in the online version, shows the spatiotemporal characteristics for the isotherms and vortex structure. The animations run over 100 rotation periods at a rate of 4 frames/s, with each frame being one1 rotation period apart (enhanced online). [URL: http://dx.doi.org/10.1063/1.3505025.7]10.1063/1.3505025.7

Image of FIG. 9.
FIG. 9.

NEU: axial vorticity ; there are seven positive (solid/black) and seven negative (white/dashed) linearly spaced contour levels in the range . Movies 9, available in the online version, show the spatiotemporal characteristics for the axial vorticity at different -levels (enhanced online). [URL: http://dx.doi.org/10.1063/1.3505025.8] [URL: http://dx.doi.org/10.1063/1.3505025.9] [URL: http://dx.doi.org/10.1063/1.3505025.10]10.1063/1.3505025.810.1063/1.3505025.910.1063/1.3505025.10

Image of FIG. 10.
FIG. 10.

NEU: time evolution of different energy modes (—); (○); (− −); ; and (◇).

Image of FIG. 11.
FIG. 11.

NEU: (a) time evolution of (a) -terms in the rate of change of kinetic energy of azimuthal perturbations, barotropic term (—), baroclinic term , centrifugal term (− −), and viscous dissipation term ; (b) the rate of change of kinetic energy ; and (c) the kinetic energy .

Image of FIG. 12.
FIG. 12.

DIR: isotherms on the planes ; at , there are 32 linearly spaced contour levels in the range , showing the formation of the corner regions and the motion after the vortex core is formed. The region shown is and . Movie 12, available in the online version, shows the spatiotemporal characteristics for the isotherms on the planes over 145 rotation periods at a rate of 10 frames/s, with each frame being one rotation period apart (enhanced online). [URL: http://dx.doi.org/10.1063/1.3505025.11]10.1063/1.3505025.11

Image of FIG. 13.
FIG. 13.

DIR: vertical profiles of the radial velocity and temperature at different radial locations and . (a) Radial velocity at times (○), 3 (◻), 7 (▽), and 20 (△). Note that the solid line indicates the depth of the Ekman layer. [(b)–(d)] Temperature profiles at (− −), 3 (○), 20 (△), 45 (◇), and 120 (×).

Image of FIG. 14.
FIG. 14.

DIR: temperature and vortex structure identified by the isosurfaces of colored by temperature. The figures on the left correspond to and on the middle/right to the vortex structure. The region shown is and . Movie 14, available in the online version, shows the spatiotemporal characteristics for the isotherms and vortex structure. The animations run over 100 rotation periods at a rate of 4 frames/s, with each frame being one rotation period apart (enhanced online). [URL: http://dx.doi.org/10.1063/1.3505025.12]10.1063/1.3505025.12

Image of FIG. 15.
FIG. 15.

DIR: axial vorticity ; there are seven positive (solid/black) and seven negative (white/dashed) linearly spaced contour levels in the range . The region shown is and . Movies 15, available in the online version, show the spatiotemporal characteristics for the axial vorticity at different -levels (enhanced online). [URL: http://dx.doi.org/10.1063/1.3505025.13] [URL: http://dx.doi.org/10.1063/1.3505025.14] [URL: http://dx.doi.org/10.1063/1.3505025.15]10.1063/1.3505025.1310.1063/1.3505025.1410.1063/1.3505025.15

Image of FIG. 16.
FIG. 16.

DIR: time evolution of different energy modes (—); (○); (− −); ; and (◇).

Image of FIG. 17.
FIG. 17.

DIR: time evolution of (a) -terms in the rate of change of kinetic energy of azimuthal perturbations, barotropic term (—), baroclinic term , centrifugal term (− −), and viscous dissipation term ; (b) the rate of change of kinetic energy ; and (c) the kinetic energy .

Image of FIG. 18.
FIG. 18.

Isotherms (top) and isobars (bottom) on the planes at . Initially , there are 15 linearly spaced contour levels in the range . The region shown is and . The symbol ● indicates a reference point for comparing the angle formed by the isotherms and isobars at that particular location.

Tables

Generic image for table
Table I.

Parameters used in the simulations. The dash in the table implies that the number from the previous column did not change.

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/content/aip/journal/pof2/22/11/10.1063/1.3505025
2010-11-08
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Numerical simulations of nonlinear thermally stratified spin-up in a circular cylinder
http://aip.metastore.ingenta.com/content/aip/journal/pof2/22/11/10.1063/1.3505025
10.1063/1.3505025
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