Effects of manifolds and corner singularities on stretching in chaotic cavity flows

It has been assumed that the stretching field in chaotic flows evolves as the result of a random multiplicative process [F.J. Muzzio, C. Meneveau, P.D. Swanson and J.M. Ottino, Scaling and multifractal properties of mixing in chaotic flows, Phys. Fluids A, 4, 1439–1456, (1992); F. J. Muzzio, P.D. Swanson and J.M. Ottino, Partially mixed structures produced by multiplicative stretching in chaotic flows, Int. J. Bifurc. Chaos, 2, 37–50 (1992)]. This assumption has been used to derive an asymptotic scaling formalism of distributions of stretching values that has useful predictive capabilities. Deviations from this scaling were thought to be limited to cases with regular islands. However, as is shown in this paper for the chaotic cavity flow, deviations from the proposed scaling can also occur for globally chaotic flows as a result of the joint action of unstable manifolds of hyperbolic periodic point and of singularities at the corners of the cavity. A detailed examination of random multiplicative stretching, the conditions necessary for its validity, and the intensity of manifold interaction effects is performed here for the cavity flow.