4. Results

4.1 Time Evolution of an Unstable Elliptical Vortex
4.2 Analysis on the Streamfunction
4.3 Visualization with Passive Contours
4.4 Finite-Time Lyapunov Analysis on Fluid Deformation
4.5 Check by Passive Contour Advection

4.5 Check by Passive Contour Advection

To check the results of finite-time Lyapunov analysis in section 4.4 , passive contours are advected as in section 4.3 so that stretching of fluid particles can be understood intuitively.

Initial conditions are given as follows:

Vortex patch: same as in section 4.1 ( thick line ).
Passive contour: eight circular contours with the radius 0.1 , centers of which are on x = 0, y = 0.45, 0.65,..., and 1.85 ( blue to red ).

Animation 3: Advection of passive contours (GIF animation; 339kB).
The contour positions are displayed from t = 0.0 to t = 10.0 at an interval of 0.1 . Thick line is the vortex patch.

- As early as t = 4.0 , the yellow and green circles are stretched out, corresponding to the areas with large Lyapunov exponents in Fig. 4 .

- Blue and light blue contours keep their shapes corresponding to the areas with small Lyapunov exponents.

- In the co-rotating frame with the ellipse, fluids in these areas are rotating oppositely from the vortex, which visualizes the clockwise flow in Fig. 5 .

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