Both pairs of vortices are observed in the frigate experiments that had been performed at MARIN. The MARIN measurements are compared to two types of large-eddy simulations. The first type uses nonlinear free-surface boundary conditions and the second type uses a low Froude-number approximation to model free-surface roughness. The subgrid-scale (SGS) turbulence model uses the Smagorinsky model. Forcing at low wavenumbers is used to balance the dissipation at high wavenumbers to provide the proper dissipation rate based on experimental measurements. The free-surface code uses an adjustment scheme to prevent the generation of spurious high-frequency waves (see Dommermuth, 1994b). The free-surface code also implements a free-surface boundary-layer approximation that is valid for clean free surfaces (see Dommermuth, 1994a).
The LES codes are initialized using the experimental data. On a transverse cut all three velocity components are measured. The mean axial and transverse velocity components are used to assign the mean velocity in a cross section of fluid that is modeled using LES. LES is used to model the turbulent portion of the wake. The Kelvin wave disturbance is subtracted out of the measured data using linear wave-cut analysis. We superimpose on top of the mean velocity field a fluctuating velocity field that is based on the rms velocity fluctuations that are measured in the towing tank. The fluctuating velocity field is initialized using a realization of fully-developed turbulence with a –5/3 inertial range. We assume that the temporal behavior predicted in our numerical results is related to the spatial behavior measured in the laboratory experiments through a Galilean transformation. The numerical predictions of the evolution of the wake correlate well with towing-tank experiments.
Consider the unsteady incompressible flow of a Newtonian fluid under a free surface, and let ui(x,t)=(u,v,w) represent the three-dimensional velocity field as a function of time. Applying Helmholtz's decomposition gives
where (x,y,z,t) is a velocity-potential which describes the irrotational flow and Ui=(U,V,W) is a solenoidal-field which describes the vortical flow such that
where ∇2 is the Laplace operator. Since satisfies Laplace's equation and the divergence of the rotational field U is chosen zero, the total velocity field u conserves mass. The potential models the effects of waves, whereas the solenoidal velocity field U models the effects of shear currents and turbulence.
Based on this Helmholtz decomposition of the velocity field, the total-pressure Π is defined in terms of a vortical pressure P and an irrotational pressure, as follows:
Here, the velocity and pressure terms are respectively normalized by uc and where uc is the characteristic velocity and ρ is the density. is the Froude number and ℓc is the characteristic length. The vertical coordinate z is positive upward, and the origin is located at the mean free surface. Substituting these decompositions (1 & 4) into the Navier-Stokes equations gives