Numerical simulation of Faraday waves

TL;DR

This paper presents a detailed numerical simulation of Faraday waves in three dimensions, accurately capturing pattern formations and comparing results with linear theory and experimental data.

Contribution

The study introduces a comprehensive 3D numerical approach for simulating Faraday waves, validating results against linear analysis and experiments, and reproducing complex pattern spectra.

Findings

Reproduced critical accelerations and wavenumbers consistent with linear Floquet analysis.

Successfully simulated and matched experimental spatiotemporal patterns.

Demonstrated detailed pattern formation including square and hexagonal structures.

Abstract

We simulate numerically the full dynamics of Faraday waves in three dimensions for two incompressible and immiscible viscous fluids. The Navier-Stokes equations are solved using a finite-difference projection method coupled with a front-tracking method for the interface between the two fluids. The domain of calculation is periodic in the horizontal directions and bounded in the vertical direction by two rigid horizontal plates. The critical accelerations and wavenumbers, as well as the temporal behaviour at onset are compared with the results of the linear Floquet analysis of Kumar and Tuckerman [J. Fluid Mech. 279, 49-68 (1994)]. The finite amplitude results are compared with the experiments of Kityk et al. [Phys. Rev. E 72, 036209 (2005)]. In particular we reproduce the detailed spatiotemporal spectrum of both square and hexagonal patterns within experimental uncertainty.