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In this paper we describe numerical studies of two dimensional turbulence near a free surface, that is a line in two dimensions, using a turbulence algorithm based on Smoothed Particle Hydrodynamics (SPH). This algorithm has been applied successfully to both decaying and driven turbulence within a two dimensional rectangular tank with rigid no-slip boundaries. The turbulence is driven by the motion of a cylinder (a disk in two dimensions) on a Lissajous trajectory. With the inclusion of gravity and a free surface, the frequencies of the disk driving the turbulence can be chosen sufficiently close to the sloshing frequency that the perturbations to the free surface are large (typically 16% of the depth). We show that, in this case, there are large differences in the velocity distribution functions and the Enstrophy near a free surface compared to those near a rigid surface. The changes in the bottom half of the tank are negligible.
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