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Singularity formation and nonlinear evolution of a viscous vortex sheet model
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10.1063/1.4789460
/content/aip/journal/pof2/25/1/10.1063/1.4789460
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/1/10.1063/1.4789460
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Evolution of the sheet for Re = ∞. Times are = 0, 0.1, 0.2, 0.3, and 0.363. The number of points is = 512.

Image of FIG. 2.
FIG. 2.

Curvature and vortex sheet strength for Re = ∞. The curvature is plotted from = 0.351 to 0.363 in the increment of 0.003. The vortex sheet strength is plotted at = 0.32, 0.34, 0.352, 0.36, and 0.363.

Image of FIG. 3.
FIG. 3.

Evolution of the sheets for Re = 1000 and Re = 200. Times are = 0, 0.1, 0.2, 0.3, 0.389 for Re = 1000, and = 0, 0.1, 0.2, 0.3, 0.438 for Re = 200. The number of points is = 1024.

Image of FIG. 4.
FIG. 4.

Curvature and vortex sheet strength of the sheets for Re = 1000 and Re = 200. (a) Curvature for Re = 1000 from = 0.377 to 0.389 in the increment of 0.003. (b) Curvature for Re = 200 from = 0.418 to 0.438 in the increment of 0.004. (c) Vortex sheet strength for Re = 1000 at = 0.34, 0.36, 0.37, 0.38, 0.386 and 0.389. (d) Vortex sheet strength for Re = 200 at = 0.38, 0.4, 0.416, 0.428, 0.434, and 0.438.

Image of FIG. 5.
FIG. 5.

Log-linear plot of the Fourier amplitudes for Re = 1000.

Image of FIG. 6.
FIG. 6.

Fits to the exponents α and β using Ansatz (16) near the singularity times for Re = ∞, 1000, and 200. The dashed lines in the β plots are at 2.57 for Re = ∞, and are at 2.5 for Re = 1000 and 200. The number of points is = 512 for Re = ∞, and = 1024 for Re = 1000 and 200.

Image of FIG. 7.
FIG. 7.

Estimation of the singularity time from the fits to α for Re = ∞ and Re = 1000. The circles are the values of α at = 30 for Re = ∞, and at = 200 for Re = 1000. The curves represent the quadratic polynomial fittings, which give = 0.364 for Re = ∞, and = 0.391 for Re = 1000.

Image of FIG. 8.
FIG. 8.

Fits to α and β for several values of near the singularity times, for Re = 3000, 1000, and 200. The time for each plot is = 0.376 for α and = 0.374 for β for Re = 3000, = 0.389 for α and = 0.386 for β for Re = 1000, and = 0.438 for α and = 0.435 for β for Re = 200. The dashed lines in the β plots are at 2.5.

Image of FIG. 9.
FIG. 9.

Fits to α and β for several values of the filter level ξ near the singularity time, for Re = 1000. (a) α for ξ = 10, (b) α for ξ = 10, (c) α for ξ = 10, and (d) β for ξ = 10, 10, and 10, at = 0.386. The number of points is = 1024 for all the cases.

Image of FIG. 10.
FIG. 10.

Dependence of the singularity time on (a) the Reynolds number and (b) the initial perturbation amplitude for Re = ∞, 1000, and 200. The dashed line in (a) is the singularity time for the inviscid case.

Image of FIG. 11.
FIG. 11.

(a) for = 128, 256, and 512, near the singularity time for Re = 1000. The solid lines are at = 0.384, 0.386, 0.388, and 0.390, and the dashed line is at = 0.391. (b) Close-up of the sheets around the center at = 0.391 for = 256, 512, and 1024.

Image of FIG. 12.
FIG. 12.

Evolution of the sheets for (a) Re = 1000 and (b) Re = 200, with δ = 0.1. Times are = 0, 0.5, 1, 1.5, and 2 for both cases.

Image of FIG. 13.
FIG. 13.

Comparison of the sheets at = 1.5, for varying Reynolds number and δ. The left plots correspond to δ = 0.15, and the right plots to δ = 0.1.

Image of FIG. 14.
FIG. 14.

Profiles of the sheets at = 2 for several values of the regularization parameter and the Reynolds number, for the initial amplitude ε = 0.05. The upper row corresponds to Re =1000, and the lower row to Re =10000. From the left column to the right, δ = 0.316, 0.158, and 0.079.

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/content/aip/journal/pof2/25/1/10.1063/1.4789460
2013-01-30
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Singularity formation and nonlinear evolution of a viscous vortex sheet model
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/1/10.1063/1.4789460
10.1063/1.4789460
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