Mixing and Transport in the Kelvin-Stuart Cat Eyes Driven Flow Using the Topological Approximation Method
Transport rates for the Kelvin-Stuart Cat Eyes driven flow are calculated using the lobe transport theory of Rom-Kedar and Wiggins through application of the Topological Approximation Method (TAM) developed by Rom-Kedar. Numerical studies by Ottino (1989) and Tsega, Michaelides, and Eschenazi (2001) of the driven or perturbed flow indicated frequency dependence of the transport. One goal of the present research is to derive an analytical expression for the transport and to study its dependence upon the perturbation frequency o. The Kelvin-Stuart Cat Eyes dynamical system consists of an infinite string of equivalent vortices exhibiting a 2pi spatial periodicity in x with an unperturbed streamfunction of H( x, y) = ln(cosh y + A cos x) - ln(1+A). The driven flow has perturbation terms of a sin(o) in both the x and y directions. Lobe dynamics transport theory states that transport occurs through the transfer of turnstile lobes, and that transport rates are equal to the area of the lobes transferred. Lobes may intersect, necessitating the calculation and removal of lobe intersection areas. The TAM requires the use of a Melnikov integral function, the zeroes of which locate the lobes, and a Whisker map (Chirikov 1979), which locates lobe intersection points. An analytical expression for the Melnikov integral function is derived for the Kelvin-Stuart Cat Eyes driven flow. Using the derived analytical Melnikov integral function, derived expressions for the periods of internal and external orbits as functions of H, and the Whisker map, the Topological Approximation Method is applied to the Kelvin-Stuart driven flow to calculate transport rates for a range of frequencies from (o = 1.21971 to o = 3.27532 as the structure index L is varied from L = 2 to L = 10. Transport rates per iteration, and cumulative transport per iteration, are calculated for 100 iterations for both internal and external lobes. The transport rates exhibit strong frequency dependence in the frequency range investigated, decreasing rapidly with increase in frequency.
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