Chaotic advection in open flows

Gaussian in (2.76), and the maximum is at A = A°° + A that gives a contour growth rate 

A special case of chaotic advection occurs in open flows in which the time-dependence of the flow is restricted to a bounded region (Tel et al., 2005). This kind of flow structure with an unsteady mixing region and simple time-independent inflow and outflow regions is typical for example in stirred reactors or in a flow formed in the wake of an obstacle. A well known example is the von Karman vortex street behind a cylinder at moderate Reynolds numbers (Jung et al., 1993 Ziemniak et al., 1994), where around the cylinder the flow is time-periodic, but at some distance from it upstream or downstream the velocity field is time independent. 

Since the time-dependence of the velocity field is restricted to a finite region the complex chaotic orbits are also limited to this region. Advection in such flows is a chaotic scattering process (Ott and Tel, 1993 Ott, 1993) in which fluid elements approach the mixing zone along the inflow streamlines, they follow chaotic trajectories inside 

Just like an isolated saddle point, the chaotic saddle has stable and unstable manifolds. The stable manifold of the chaotic saddle is a 

The chaotic saddle and its manifolds are also sets of zero measure with fractal structure. The set of points, seen in Fig. 2.13 corresponding to inflow coordinates with very large, singular, escape times, typically form also a fractal set determined by the intersection of the saddle s stable manifold and the line containing the initial conditions. There is a connection between the dimension of the chaotic saddle and the dimensions of its manifolds. The trajectories on the chaotic saddle have a set of Lyapunov exponents whose number is equal to the dimension of the full space, d. The sum of the Lyapunov exponents is zero due to incompressibility and chaotic dynamics implies 

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