|
Items in cart [0] |
Questions? Call 888-624-8373 |
|
||||||||||||||||
|
||||||||||||||||
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. of the local models improve. All of the SGS coefficients increase as coarseness increases. The Type I model with Δe/Δ=16/5 roughly corresponds to one case that Lund and Novikov investigated using a spectral code with a sharp-cutoff filter and volume averaging [13]. For this case their correlation coefficient (C=0.24) agrees with our results. As a possible explanation of the excellent correlation coefficients for the global model, consider the correlation between the energy density (ukuk) and the filtered energy-density 6.2A Posteriori TestsTable 8 in Section 9 at the end of this paper provides the details of six DNS studies of free-surface turbulence. These DNS studies, which include simulations of subsurface turbulence without waves, turbulence with waves, and waves without turbulence, are compared to low-resolution LES. The LES studies and the DNS studies that they are compared to are provided in Table 9. The initial assignment of the subsurface-velocity field is similar to the procedure Dommermuth (1993a) discusses in his Appendix C [ 6]. The initial shape of the subsurface-velocity spectrum is κ−5/3. The mean velocity components are zero. To ensure that the LES and DNS studies use identical initial conditions, only low wavenumbers (κ≤12π) that will fit into the LES are excited. Free-slip boundary conditions are initially used on the plane z=0. If a surface wave is also present, then the subsurface-velocity field is periodically extended above the plane z=0. A boundary-value problem is solved to set the normal-component of the surface velocity to zero (see Equation 8). The initial rms velocity of the subsurface-velocity field is set to one, whether or not a surface wave is present. The surface wave is assigned using an exact Gravity wave, and the generation of spurious high-frequency waves due to imbalances in the initial conditions is eliminated using an adjustment procedure (see [6], [7], & [8]). The simulations are run for about 2.5 wave periods, or more than two small-scale eddy-turnover times (to) for the lowest Froude-number runs and almost six turnover times for the highest Froude number, where to=λo/uo, λo is the final Taylor microscale, and uo is the final rms velocity. The spectrum of the total-velocity field is calculated by taking the Fourier transform in the horizontal plane of Figures 4 & 5 compare the spectra of the DNS to the LES for cases with and without surface waves. Both figures include the spectrum of the total velocity field (E(κ)) and the spectrum of the wave energy (S(κ)). By comparing the initial velocity spectra to the final velocity spectra (see Figures 4a & 5a), we observe that the higher wavenumbers (κ>12π) of the DNS runs have filled in according to Kolmogorov's law (E(κ) ∝ κ−5/3). The spectra of wave energy in Figure 4b show a buildup of energy for wavenumbers κ>2π relative to the initial conditions. This buildup corresponds to the formation of parasitic capillary waves on the front face of the
|
|
|||||||||||||||
. For even the coarsest filter, this correlation exceeds 0.97. This implies that most of the kinetic energy is concentrated in the lowest wavenumbers, which helps to explain the excellent correlation of the global model. Moreover, as noted by Meneveau, et al (1992), although the global model captures a significant portion of the SGS energy, its wavenumber content is too high to be resolved by a LES formulation [
where D is the depth and η+D accounts for the vertical extent of the fluid. The Fourier coefficients are squared and integrated over the depth. Finally, the energy density is summed over wavenumber shells to calculate the one-dimensional spectrum E(κ). Details of a similar procedure are provided in Dommermuth (1993a).