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4.3. NUMERICAL METHODS 29 where k is the turbulent kinetic energy and Ui is the mean flow velocity components. To close this equation the turbulent viscosity, υT has to be ex- pressed in terms of the mean quantities. The most common turbulence models are the two equations models, which are based on transport equations. One of these models is the k-ε model, and in this model, the transport equations are solved for the turbulent kinetic energy and its dissipation ε. From these two quantities, the turbulent viscosity can be determined by the relation: k2 υT =Cμ ε (4.9) Cμ is a model constant. There are also other alternatives of the stan- dard k-ε model, for example the Renormalisation Group (RNG) k-ε models, which has an additional term in the ε-equation to take into account the ef- fects of the mean flow on distortion of the turbulence. This model has been used for all the RANS computations presented in this thesis. In the Reynolds- stress models, model transport equations are solved for the Reynolds stresses, uiuj and for the dissipation, ε. Since the Reynolds stresses are known, the turbulent-viscosity hypothesis is not needed. This has the advantage that ef- fects of anisotropy of the turbulence, streamline curvature, swirling motions and high rate of strains are taken into account. Still, some terms in the Rey- nolds stress transport equations have to be modelled, with some more or less realistic assumptions. Unfortunately, there are no generally valid turbulence models. For plane boundary layer flow, most turbulence models work more or less well, simply since such simple flows are used for model calibration. On the other hand, the flow near a solid wall depends strongly on the shape of the boundary and the flow outside of the boundary layer. These effects make it difficult to have generally valid models. Treatment of the near wall turbulence is often done by using certain wall models. These models may work when the local flow conditions resemble those for which the model is calibrated for. The most common wall model, is the so called (standard) wall function, which is based on the log-law behaviour of the turbulent boundary layer on a plane wall at zero pressure gradient. Modifications to the logarithmic law of the wall to take into account effects of wall curvature and adverse pressure gradients have also been proposed. A more advanced wall model, is the so called two-layer model, where simplified turbulence models are used in the viscous region close to walls whereas high Re turbulence model is used further away from the wall. In LES, the large scales of the turbulent flow are resolved while the smaller scales are modelled. This implies that the resolved flow field is, in general, 3- dimensional and time dependent. These properties of LES makes it a natural candidate for handling the pulsatile flow in the turbine. In LES, the dependent variables are low-pass filtered with a spatial filter. The filter width is defined through the smallest spatial resolution which can be related to the volume ofPDF Image | Numerical computations of the unsteady flow in a radial turbine
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