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The following HTML text is provided to enhance online readability. Many aspects of typography translate only awkwardly to HTML. Please use the page image as the authoritative form to ensure accuracy. steepness and the shear. The error in the surface vorticity ( 5.5Test Case V: Gerstner's WaveTable 5 shows the numerical errors as a function of the spatial and temporal resolution and the nonlinearity when Gerstner's wave is used as initial conditions[12]. A Newton-Raphson technique is used to convert Gerstner's Lagrangian coordinates to our Eulerian coordinates. Then a boundary-value problem is solved to convert from the primitive-variable formulation of Gerstner to our Helmholtz decomposition. The characteristic length is the wavelength ℓc=λ, and the characteristic velocity is Table 5: Gerstner's wave convergence test.
The wave steepness, The errors in the surface vorticity show that the fourth-order accuracy is approached as the grid resolution is increased. However, the convergence of the free-surface solution is not as rapid. A closer inspection of the numerical results shows that the cusp that occurs at the crest of the Gerstner wave is smoothed out. Another factor that contributes to the slower convergence is the finite-depth effects that are present in the numerical solution, but not the analytic solution. Even with these limitations, the accuracy of these numerical simulations is comparable to the simpler gravity-wave case. 6LES PERFORMANCE STUDIESThe LES formulation is tested a priori using a DNS dataset of three-dimensional homogeneous turbulence and a posteriori using numerical simulations of free-surface turbulence. The DNS code that is used to generate the dataset of homogeneous turbulence uses the same finite-difference operators and multigrid solver as the free-surface code. The study of homogeneous turbulence allows us to perform high-resolution simulations of a flow that is less complex than free-surface turbulence. The LES studies of free-surface turbulence include comparisons to moderate-resolution DNS studies of turbulence without waves, turbulence with waves, and waves without turbulence. These numerical simulations allow us to assess the performance of the test-filter approach and the SGS closures under a variety of conditions. 6.1A Priori TestsThe a priori tests of the SGS models are based on a DNS simulation of homogeneous turbulence. Based on a length scale of ℓc= 5cm and a rms-velocity scale of uc=4cm/s, the Reynolds number is Re=2000. These scales are chosen to match our free-surface turbulence studies, which we will discuss later. As initial conditions, we used white noise with zero mean that was projected onto a solenoidal velocity field. A 1283 numerical simulation was run until the kinetic energy decayed to about 25% of its initial value. Then the velocity field was rescaled so that
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ω) is normalized by 
= |U1|π/2h. The errors in the free-surface elevation are halved when the magnitude of the shear is halved. Similarly, the errors in the surface vorticity are halved when the wave steepness is halved. The kinetic and potential energy is conserved to within .05% for all cases.
. The depth is equal to the wavelength. The solution is inviscid and rotational. Capillarity is not modeled.
, is a function of zs: 
, where k is the wavenumber and zs=0 corresponds to a cycloid. The two entries in the table for zs=−0.1 and −0.2 correspond to 
, where Ω is the wave frequency.