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30 4. METHODS the computational cell, V 1/3, i.e. ∆=V 1/3. The governing equations for the filtered variables have additional terms as compared to the original equations. These terms have to be expressed in terms of the filtered variables in order to close the equations. In the momentum equation, the term that has to be modelled is called the Sub-Grid-Scale stress (SGS-model). As for the RANS approach, different SGS-models exist, and the first proposed is the Smagorinsky model. In the Smagorinsky model, as in the RANS framework one introduces a modified viscosity and assumes that the effects of unresolved turbulence can be accounted for in analogy with the molecular viscosity. Thus, the total viscosity is the sum of molecular and SGS-viscosities: υ = υP hysical + υSGS . υSGS is assumed to be proportional to the absolute value of the rate of strain tensor S ̄ij and the filter width: υSGS =(Cs·∆)2S ̄ij (4.10) where Cs is a model constant. It has long been known that the Smagorin- sky model is dissipative, especially in the near wall regions and when rotational effects are present. In analogy with the RANS-approach, one has also proposed a one equation eddy viscosity model, where a transport equation for the unre- solved turbulent kinetic energy is solved. In this model, υSGS is assumed to be proportional to the filter width and the unresolved turbulent kinetic energy: υSGS = Ck · ∆ · kSGS (4.11) One may argue against this model in addition to the simple fact that it has similar limitations as the Smagorinsky model. The small scales of turbu- lence are of local and universal character (i.e. independent on the particular problem due to their small scale, except close to the wall). Therefore the SGS effects should be expressed by local variables and not through an elliptic partial differential equation. Additionally, one may argue that the eddies of different scales interact with each other similarly if the scale ratio is similar. This is the foundation of several SGS models such as the scale similarity and dynamic models. In the SGS models based on the scale similarity hypothesis, the subgrid tensor is approximated by an analogous tensor computed from the smallest resolved scales, c.f. Bardina et al. (1980). In the dynamic models, the SGS model coefficients are based on the local properties of the flow field, which implies that the coefficients are functions of space and time, c.f. Ger- mano et al. (1991) and Ghosal et al. (1995). Based on the same argument one may also claim the error in neglecting the SGS terms is of second order in the filter size. Thus, if the spatial resolution of the flow is high enough the SGS terms are small. In fact as the resolution is improved LES becomes DNS. With this in mind one may claim that for adequate resolution one may refrain cell cellPDF Image | Numerical computations of the unsteady flow in a radial turbine
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