Numerical simulation of a two-frequency-driven superlattice Faraday-wave pattern
The formation of a superlattice pattern in two-frequency-driven Faraday waves discovered and named SSS-I by Arbell & Fineberg (1998, 2002) is investigated by means of Direct Numerical Simulations (DNS) of the full three-dimensional Navier--Stokes equations with a free surface. Two simulations with distinct quasi-hexagonal initial conditions run at a forcing amplitude $25\%$ above the Faraday-wave onset followed quite different routes, but both led eventually to the same superlattice pattern after around 250 forcing periods. This regime is inaccessible to the approximations of weak nonlinearity or viscosity. The standing-wave pattern contain rows of patches, alternating in time between hills and lakes that are connected by a long skeleton resembing the backbone of DNA strands. The patches and skeleton of the pattern can be related to its spatial Fourier decomposition, which combines hexagonal modes with a spatially and temporally subharmonic mode. One of the transition routes passes through several fairly long-lived transients including different hexagonal patterns and another superlattice pattern; the other passes only through erratic and disordered states. After another 100 periods, the pattern became unstable and was succeeded by a dynamic version of SSS-I in which the superlattice is modulated and drifts in the direction of the backbone, while preserving its basic shape. Convergence to SSS-I states both experimentally in a large geometry and numerically from two different initial conditions and in a minimal geometry demonstrates the robustness of the SSS-I pattern.