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4.3. NUMERICAL METHODS 31 from using an explicit expression for the SGS terms. This is the foundation for the so called Implicit LES, sometimes called ILES or MILES=Monotone Implicit LES. The size of the SGS term is only one factor in the considerations related to the modelling of the SGS terms. The SGS terms represent the effect of the small scales on the large ones. They should account for a net energy transfer from the resolved scales to the unresolved dissipative ones. Instanta- neously, they also act to transfer energy in the other direction, an effect that is known and backscatter and which has not been quantified. Since the small- est scales are unresolved the SGS model should account for the dissipation. When no explicit SGS model is used, one may utilize the inherent dissipative properties of the numerical scheme, provided that the dissipation of the larger scales is negligible. It should be, however, emphasized that the accuracy of these methods depends of the grid size and numerical schemes. A first order scheme implies that the cut-off frequency will be lower compared to a higher order scheme since the artificial viscosity (dissipation) is higher for first order schemes compared to second order schemes. But, for seconds order schemes, such as the central difference scheme, the odd derivate of the truncation error can introduce dispersion, which can result in non-physical oscillations in the flow field. The effect of these factors on the resolved scales is often small pro- vided that the grid resolution is fine as compared to the resolved scales that are of interest. Marginally resolved flows may of course suffer from lack of accuracy since the numerical dissipation is proportional to the grid spacing. A measure of the local resolution can be assessed by considering the local turbulent energy spectrum and the extent of the resolved inertial subrange. A disadvantage with LES is that the boundary layers have to be resolved to capture the dynamics in the near-wall regions. To be able to do this, the near-wall region must be resolved which results in a very large number of grid points. As in the RANS case, different wall models have been proposed and these can be divided into two categories: • Modifying the SGS eddy viscosity in the region closest to the wall, for example a damping function of the viscosity or compute the eddy viscosity based on the logarithmic law of wall. • Solve the one-dimensional boundary layer NS equations on a fine grid in the near-wall region. The fine grid can also be a “grid-within-the grid”, where a one-dimensional grid is included in the cell layer closest to the wall. In this thesis, the Implicit LES approach is used for almost all cases, and hence, no near-wall model is used at all. For the case where the Smagorinsky model is used, a damping function is applied at the cell layer adjacent to the wall, where the filter width in this cell layer is defined as:PDF Image | Numerical computations of the unsteady flow in a radial turbine
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