Experimental measurement of the Melnikov function

Abstract

International audienceWe study the transport properties of a genuine two-dimensional flow with a large mean velocity perturbed periodically in time by means of an original experimental technique. The flow, generated by the co-rotation of two cylinders is both stratified with a linear density gradient using salted water, and viscous in order to prevent Ekman pumping and centrifugal instabilities. Thus, the mean flow contains a hy-perbolic point with a homoclinic streamline, which we perturb periodically by an extra oscillation. A blob of scalar injected close to the stagnation point contracts on the stable manifold, and stretches in the unstable direction. The distance between the stable and the unstable manifolds is measured as the distance between the maximum and the minimum of the dye undulating pattern, and is recorded as a function of the perturbation frequency. This distance, also called the Melnikov function, presents a maximum when the residence time of a fluid particle in the mean flow is about half a perturbation period. This resonance criterion is recovered with good quantitative agreement by the theoretical prediction of the Melnikov function computed for this flow

Chaotic mixing in an acoustically driven cavity flow

“…The matter further triggered a great amount of theoretical, numerical, or experimental works on the subject (see, for instance, Refs. [25][26][27][28][29][30][31][32][33]). Note finally that chaotic advection also applies for dynamical systems of higher dimension, as, for instance, 3D time-periodic flows [34][35][36][37][38].…”