Numerical computations of the unsteady flow in a radial turbine

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Numerical computations of the unsteady flow in a radial turbine ( numerical-computations-unsteady-flow-a-radial-turbine )

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28 4. METHODS it particularly well suited to capture the strong gradients expected for these cases. Since the MARS scheme is proprietary to CD-adapco, no details of the scheme can be found in the open literature. According to CD-adapco (2005) this scheme is the scheme that is least sensitive to solution accuracy to mesh structure and skewness of the available schemes in the used code. Since the details of the scheme are not known we have made extensive tests to assess its accuracy for problems of relevance to engine flows. A modified version of the Pressure-Implicit with Splitting of Operators (PISO) method proposed in Issa (1986), Issa et al. (1986) and Issa et al. (1991) is implemented in the used code to solve the discretized governing equations at each time step. One of the major differences is that the number of corrector stages is not limited to two, as being proposed in the original version; instead, the number of corrector stages is determined by the splitting error, which will, according to CD-adapco (2005), increase the accuracy and reliability of the method. 4.3.1. Turbulence modelling To resolve all scales of the turbulent flow the grid size and the time step must be smaller than the smallest length and time scale of the flow. This can be done by fully resolved simulations, Direct Numerical Simulation, and in general, the computation cost will be proportional to Re3. This approach is not feasible for the applications studied in this work, since the required computational resources are not available yet. Instead, the governing equations can be handled by two other methods, the Reynolds Averaged Navier Stokes (RANS) approach and the Large Eddy Simulation (LES) approach. In the RANS approach, the governing equations are expressed in terms of the mean quantities. The base line of the RANS equations is that the instantaneous flow field can be divided into a time averaged part and a fluctuating part. With this inserted in the governing equations and averaged over time, the RANS equations are obtained, which describes the time averaged flow field. Due to the non-linearity 2 􏰘∂Ui ∂Uj 􏰙 uiuj = 3kδij −υT ∂x + ∂x (4.8) ji of the NS equations, additional terms appear in the RANS equations; the ′ ̄ ′ Reynolds stresses, uiuj. These terms represent the effect of the fluctuations on the mean field. The Reynolds stresses cannot be expressed analytically in terms of the mean variables and hence they have to be modelled in order to close the equations. Turbulence models can be divided in several major categories, such as Eddy viscosity models and Reynolds stress based models. The eddy viscosity models are based on the turbulent-viscosity hypothesis which relates the Reynolds stresses to the mean velocity field as: ′ ̄ ′

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