|5. Concluding Remarks|
The time evolution of an unstable elliptical vortex patch was analyzed using a contour dynamics model.
Calculations of streamfunctions, passive contour advections, and finite-time Lyapunov exponents revealed the deformation mechanism of the vortex patch and gave visualized recognitions and a quantitative measure of the fluid mixing.
Deformation of the Elliptical Vortex Patch
- For the perturbation of m = 3 (antisymmetric) , a streamer is formed from one edge of the elliptical vortex and largely deformed ( Anim. 1 ) .
- In the co-rotating frame with the vortex patch ( Fig. 2 ), the streamlines containing a stagnation point approach the vortex contour, and after that, the stagnation point comes into the vortex, which causes the streamer formation.
- Any remarkable difference cannot be recognized at the initial state between the both edge of the vortex in any analyzes, except a slight asymmetry in the streamfunction.
Visualization of the Fluid Mixing
- In the contour dynamics, only the contours of vorticity jump are paid attention and other regions tend to be missed. However, the passive contour advection show the complex flow in the regions other than the vortex contour ( Anim. 2, Anim. 3 ) .
- The mixing is characterized through the flow field in the co-rotating frame. The passive contours are deformed near the stagnation points, and stretched along one of the streamlines containing the stagnation point ( Fig. 2 ).
- The inside of the vortex patch is separated from the outer region, and there is no mixing between these two regions.
- Two regions exist slightly outside of the minor axis of the ellipse that are surrounded by the streamlines containing the stagnation points, in which the flow is clockwise ( Fig. 5 ). This region is also separated from the other regions with little mixing until the streamer formation.
Quantitative Measure of the Fluid Mixing
- Finite-time Lyapunov exponents give a quantitative measure of the deformation at each point ( Fig. 4 ).
- The regions of large Lyapunov exponents, which are near the streamlines containing the stagnation points, are extended from the edge of the ellipse as the evaluation time increases.
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