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Unsteady flow separation on slip boundaries
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10.1063/1.2923193
/content/aip/journal/pof2/20/9/10.1063/1.2923193
http://aip.metastore.ingenta.com/content/aip/journal/pof2/20/9/10.1063/1.2923193

Figures

Image of FIG. 1.
FIG. 1.

Separation in the absence of hyperbolic stagnation points and without any free streamlines detaching from the boundary. The streamfunction is given by . For , the streamlines are closed curves (dashed lines) and there is only one elliptic stagnation point oscillating near the origin. Numerical pathlines and analytical solutions reveal flow separation from the boundary, which has neither stagnation points nor detaching streamlines. The figure corresponds to (enhanced online). [URL: http://dx.doi.org/10.1063/1.2923193.1]10.1063/1.2923193.1

Image of FIG. 2.
FIG. 2.

Separation is the breakaway of fluid near a wall. Particles in , a portion of the boundary that acts as a stable manifold, are attracted toward the separation trajectory . Fluid is ejected along an unstable manifold inside the fluid domain . A necessary and sufficient criteria for separation is derived by transforming the velocity field into a moving coordinate frame oriented with the unit tangent vector and the unit normal vector . The existence of a normally hyperbolic subspace generates the desired separation framework.

Image of FIG. 3.
FIG. 3.

Separation in periodically oscillating convection cells. The invariants of the Poincaré map of the system correspond to periodic hyperbolic trajectories on the top and bottom boundaries (triangles). The flow separates along the unstable manifold of the hyperbolic trajectory oscillating along the top boundary. Particles reattach to the lower boundary along the unstable manifold of the other hyperbolic trajectory.

Image of FIG. 4.
FIG. 4.

Comparison between the hyperbolic Poincaré invariants (classical theory) and the point corresponding to the maximum for periodic Rayleigh–Bénard convection cells. Left panel: The Poincaré section of the system has invariants that correspond to hyperbolic (separating) periodic trajectories. The maximum of is visually identical to the separation point predicted by Poincaré. Right panel: The difference between Poincaré’s separating trajectory and the point where is maximum, remains below 0.1% of the amplitude of the oscillations regardless of the integration time .

Image of FIG. 5.
FIG. 5.

Amplitude of the random forcing (which also corresponds to a stagnation point in this case) and corresponding separation and reattachment points. This chart gives only the coordinate of the points. For the separation point, the coordinate is . For the reattachment point, the coordinate is . Note that , shifted by , corresponds to the coordinates of the stagnation points both on the top and on the bottom boundaries.

Image of FIG. 6.
FIG. 6.

Transport in a chaotic Rayleigh–Bénard cell under random Gaussian forcing with amplitude and decorrelation time . Figures 6–8 show streamlines [Eq. (7) with and ], as well as separation and reattachment points at different times. Also shown are the fields in forward (below streamlines) and backward time (above streamlines). The color map is identical for the two plots and their maxima mark separation or reattachment. Particles (thick lines) released from five fixed locations near the top boundary confirm the predicted position and angle of separation. This panel corresponds to time .

Image of FIG. 7.
FIG. 7.

Transport in a chaotic Rayleigh–Bénard cell under random Gaussian forcing with amplitude and decorrelation time . Figures 6–8 show streamlines [Eq. (7) with and ], as well as separation and reattachment points at different times. Also shown are the fields in forward (below streamlines) and backward time (above streamlines). The color map is identical for the two plots and their maxima mark separation or reattachment. Particles (thick lines) released from five fixed locations near the top boundary confirm the predicted position and angle of separation. This panel corresponds to time .

Image of FIG. 8.
FIG. 8.

Transport in a chaotic Rayleigh–Bénard cell under random Gaussian forcing with amplitude and decorrelation time . Figures 6–8 show streamlines [Eq. (7) with and ], as well as separation and reattachment points at different times. Also shown are the fields in forward (below streamlines) and backward time (above streamlines). The color map is identical for the two plots and their maxima mark separation or reattachment. Particles (thick lines) released from five fixed locations near the top boundary confirm the predicted position and angle of separation. This panel corresponds to time .

Image of FIG. 9.
FIG. 9.

Robustness of the separation point with respect to model parameters. Left panel: Influence of the maximum velocity . Right panel: Influence of the amplitude of the lateral displacement .

Image of FIG. 10.
FIG. 10.

Transport in a chaotic noisy Rayleigh–Bénard cell using a random Gaussian forcing with amplitude and decorrelation time . Each panel shows the streamlines, separation, and reattachment points at time for different noise fields . (a) [URL: http://dx.doi.org/10.1063/1.2923193.2]. (b) , , [URL: http://dx.doi.org/10.1063/1.2923193.3]. (c) , , [URL: http://dx.doi.org/10.1063/1.2923193.4]. (d) , , [URL: http://dx.doi.org/10.1063/1.2923193.5]. (e) , , [URL: http://dx.doi.org/10.1063/1.2923193.6]. (f) , , [URL: http://dx.doi.org/10.1063/1.2923193.7]. Particles (thick lines) released from five fixed locations on the top boundary validate the position and angle of the separation. Labels and color map are identical to those in Figs. 6–8 (enhanced online).10.1063/1.2923193.210.1063/1.2923193.310.1063/1.2923193.410.1063/1.2923193.510.1063/1.2923193.610.1063/1.2923193.7

Image of FIG. 11.
FIG. 11.

Transport in a chaotic noisy Rayleigh–Bénard cell using a random Gaussian forcing with amplitude and decorrelation time . Each panel shows the streamlines, separation, and reattachment points at time for different noise fields . (a) (enhanced online) [URL: http://dx.doi.org/10.1063/1.2923193.8]. (b), , (enhanced online) [URL: http://dx.doi.org/10.1063/1.2923193.9]. (c), , (enhanced online) [URL: http://dx.doi.org/10.1063/1.2923193.10]. (d), , (enhanced online) [URL: http://dx.doi.org/10.1063/1.2923193.11]. (e), , (enhanced online) [URL: http://dx.doi.org/10.1063/1.2923193.12]. (f), , [URL: http://dx.doi.org/10.1063/1.2923193.13]. Particles (thick lines) released from five fixed locations on the top boundary validate the position and angle of the separation. Labels and color map are identical to those in Figs. 6–8 (enhanced online).10.1063/1.2923193.810.1063/1.2923193.910.1063/1.2923193.1010.1063/1.2923193.1110.1063/1.2923193.1210.1063/1.2923193.13

Image of FIG. 12.
FIG. 12.

Robustness of the separation point with respect to multivariate Gaussian noise.

Image of FIG. 13.
FIG. 13.

Reattachment point in Monterey Bay. Highest level sets of as a function of time and the arc length along the coastline. and are approximated using a finite integration time of . There is at most one reattachment point at each time, corresponding to a positive value of .

Image of FIG. 14.
FIG. 14.

Separation and reattachment points in Monterey Bay. The upper panels show the separation point near Santa Cruz as well as streaklines released near the coast. The panels at the bottom show, in addition, the reattachment point and streamlines obtained from a backward-time integration. In this case, the base of the reattachment profile wanders around the Monterey Peninsula.

Image of FIG. 15.
FIG. 15.

Sketch illustrating the geometry of transport in Monterey Bay based on the separation and reattachment profiles.

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/content/aip/journal/pof2/20/9/10.1063/1.2923193
2008-09-03
2014-04-25
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Unsteady flow separation on slip boundaries
http://aip.metastore.ingenta.com/content/aip/journal/pof2/20/9/10.1063/1.2923193
10.1063/1.2923193
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