Lagrangian motion of fluid particles induced by three-dimensional standing surface waves

Abstract

Lagrangian motion of fluid particles on three-dimensional weakly nonlinear standing surface waves is studied by inviscid irrotational theory. The three-dimensional flow inside the fluid in a square cylinder is derived when two internally resonant modes (m,n) and (n,m) are excited. Using the property that particles initially located on the surface continue to float on it, the three-dimensional flow can be reduced to two dimensional. It is found that a second-order drift velocity emerges on the surface, which is proportional to the angular momentum associated with the surface displacement. The drift velocity is vortical and consists of derivatives of modes interacting with the two first-order modes. Streamline patterns of drift motion show a large-scale structure for m≂n and a small-cell flow for m≫n. The surface flow is a superposition of this drift velocity and the first- and second-order oscillatory velocity with zero time average. When the expansion parameter ε remains small, the numerical Poincaré plot of particle paths for the two-dimensional surface flow shows sets of tori and heteroclinic orbits. Breakdown of tori by resonance of the wave oscillation and drift flow and the period-doubling bifurcations create chaotic layers as ε becomes large. For sufficiently large ε, a single particle moves randomly over a distance of the size of a container. Numerically computed Lyapunov exponents and fractal dimension show that the random particle motion is chaotic and weakly dissipative.