2.1 Contour Dynamics
2.2 Equations of Cotour Dynamics
2.3 Theory of an Elliptical Vortex
2.4 Finite-Time Lyapunov Analysis on Fluid Deformation
2.1 Contour DynamicsMotion in two-dimensional nondivergent perfect fluid can be described using Lagrangian conservation law of vorticity:
where u( x,t ) is the velocity, and Q( x,t ) is the vertical component of the vorticity. The vorticity distribution is assumed to be
for regions Dj ( j=1,2, ... ,J ) ( Dj1ÇDj2 = j ) on an infinite plane. Each region is called vortex patch.
In this situation,
Such dynamics concerning time evolution of the vortex patch is called contour dynamics.
- The vorticity in each region keeps constant in time due to Lagrangian conservation law.
- Thus time evolution of the system can be obtained by chasing the position of the contours (boundary lines) of the regions.
- Time evolution of the contour positions can be calculated from the contour positions themselves, because the velocity at an arbitrary point can be calculated by them.
- As a result, time evolution of the system is described as the contour deformation.
- Reducing two-dimensional problem to one-dimensional line integration decreases calculating time remarkably.
- It is easy to treat unsteady behavior.
- Reliability has been confirmed through conparative experiments with a spectral method, and it is possible to calculate in higher resolution.
- Due to the difficulty in removing the perfect fluid assumption, it is not appropriate for problems in which dissipation process is important.
- It does not have enough reality for application since it is considerably a conceptual model.
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