Two-dimensional direct numerical simulation of parametrically excited surface waves in viscous fluid

Standing surface waves on a viscous fluid driven parametrically by a vertical harmonic oscillation are investigated, based on direct numerical simulations of the two-dimensional Navier-Stokes equation, together with appropriate boundary conditions. The condition for the onset of the waves in the experiments by Lioubashevski [Phys. Rev. Lett. 76, 3959 (1996)] is reproduced by our numerical simulation. The time evolution and the flow structures are investigated in detail. The form of the surface elevation is analyzed and the dependence of the saturated amplitude on the forcing strength shows a normal bifurcation. Instead of a localized state, spatially uniform standing waves are formed in an extended system. Using initial perturbations of the uniform state, numerical simulations show that the uniform standing waves are stable to two-dimensional disturbances, which suggests that three-dimensionality is essential for the spatially localized state to occur. (C) 2001 American Institute of Physics.