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Relationship between stirring rate and Reynolds number in the chaotically advected steady flow in a container with exactly counter-rotating lids
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View: Figures


Image of FIG. 1.
FIG. 1.

(Color online) (a) Geometry and base-flow patterns for cylindrical container with exactly counter-rotating lids. (b) Typical computational grid.

Image of FIG. 2.
FIG. 2.

Grid sensitivity study ( and ). Contours of the radial (top), axial (middle), and azimuthal (bottom) velocity components at the equatorial (left column) and a diametral (right column) planes. Dashed line: grid A; thin line: grid B; thick line: grid C.

Image of FIG. 3.
FIG. 3.

(Color online) Contours of radial vorticity and surface streamlines at the azimuthal plane . The azimuthal plane has been unfolded and plotted in two dimensions (, ). (a) , ; (b) , ; (c) , ; (d) , ; (e) , .

Image of FIG. 4.
FIG. 4.

(Color online) Pressure contours and 2D streamlines at the equatorial plane. (a) , ; (b) , ; (c) , ; (d) , ; (e) , .

Image of FIG. 5.
FIG. 5.

(Color online) 3D particle paths originating just above and below of one of the three pairs of saddle foci shown in Fig. 4 (, ). Trajectories have been tracked forward and backward in time.

Image of FIG. 6.
FIG. 6.

(Color online) 3D structure of the flow (, ). (a) Vortical structures using the method . The dashed line and circle mark the axial inclined and radial vortices, respectively; (b) superposition of the contour with 2D streamlines on an plane; (c) lower half of the isosurface shown in (a) superimposed with 2D streamlines on the equatorial plane and particle paths originating from the saddle foci.

Image of FIG. 7.
FIG. 7.

(Color online) Snapshots depicting the stirring of two blobs of particles (, ). The red blob is placed initially in an invariant region while the blue blob is placed in a chaotically advected flow region.

Image of FIG. 8.
FIG. 8.

(Color online) Schematic illustrating the 2D- numerical technique for calculating the discrete concentration field. A blob of particles is placed at a location within a chaotically advected region of the flow (a). At every instant in time the instantaneous locations of all particles (b) are mapped to a single plane (c).

Image of FIG. 9.
FIG. 9.

(Color online) Instantaneous concentration maps for (, ) using the 2D- numerical technique (time is measured in lid revolutions).

Image of FIG. 10.
FIG. 10.

Calculated time series of the discrete variance of concentration using the 2D- numerical technique for and various Reynolds numbers (semilog plot).

Image of FIG. 11.
FIG. 11.

Calculated stirring rates as a function of the Reynolds numbers (). For selected Reynolds numbers the stirring rate has been calculated using various variants of the discrete VOC approach (3D, 2D-, and 2D-z) and using flow fields obtained on grids B and C.

Image of FIG. 12.
FIG. 12.

(Color online) Lagrangian time-average maps for various Reynolds numbers . The plane where 10 000 initial conditions were uniformly distributed is shown at the top.

Image of FIG. 13.
FIG. 13.

(Color online) Poincaré maps for and and 350. The plane of section is the same as that shown in Fig. 12.

Image of FIG. 14.
FIG. 14.

(Color online) Particle paths depicting unmixed regions in the flow for various Reynolds numbers . Due to symmetry the toroidal invariant (black lines) core is shown only at the lower half of the container.

Image of FIG. 15.
FIG. 15.

Variation of calculated stirring rate with Reynolds number in log-log scale . The solid line marks the rate of decay of stirring rate predicted by the theory of Mezić (Ref. 8).


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Relationship between stirring rate and Reynolds number in the chaotically advected steady flow in a container with exactly counter-rotating lids